{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0001","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"791960f8e00d450237a1059d7905ed49e5540532d3a53aaaaec8d8af59db8b71","family_id":"qf-mod34-statmodel-qualification","seed":202611034001,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,4],[1,4],[2,4]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/4, 1/4, 2/4], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/4, 1/4, 1/2], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/4, 1/4, 1/2], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0002","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90a415a4f7a1f890bd471e0e15ed75c2bd3c6cbaeab87f6852aee84a3246651c","family_id":"qf-mod34-statmodel-qualification","seed":202611034002,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,5],[1,5],[3,5]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/5, 1/5, 3/5], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/5, 1/5, 3/5], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/5, 1/5, 3/5], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0003","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c91a8c5b96b288b46dc0792e16fafa8e50ff485048bfe6a97b092a35c10390ea","family_id":"qf-mod34-statmodel-qualification","seed":202611034003,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,6],[1,6],[4,6]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/6, 1/6, 4/6], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/6, 1/6, 2/3], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/6, 1/6, 2/3], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0004","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d75e212c28324a12eb28d21c1a730e308778f97f2a978c30e8bbce119a435eb","family_id":"qf-mod34-statmodel-qualification","seed":202611034004,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,7],[1,7],[5,7]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/7, 1/7, 5/7], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/7, 1/7, 5/7], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/7, 1/7, 5/7], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0005","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"846fdf1ad1c3e9d1b55e969b32f587171c2eeb748c7b684209722e4a5e1262db","family_id":"qf-mod34-statmodel-qualification","seed":202611034005,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,8],[1,8],[6,8]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/8, 1/8, 6/8], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/8, 1/8, 3/4], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/8, 1/8, 3/4], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0006","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6010e7f34f7c754c07686a36727de86851021e5bf3e2957913cbd12dc2179833","family_id":"qf-mod34-statmodel-qualification","seed":202611034006,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,9],[1,9],[7,9]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/9, 1/9, 7/9], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/9, 1/9, 7/9], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0007","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"110c4759edacdb58e2c3c62bda2be9f8f6d9162408fc43f9ee63c0c75fd4ee4a","family_id":"qf-mod34-statmodel-qualification","seed":202611034007,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,10],[1,10],[8,10]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/10, 1/10, 8/10], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/10, 1/10, 4/5], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/10, 1/10, 4/5], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0008","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa4c97e7912b1a848846cb064a70cb6316b82db6d7b6e1374ba36310d0242509","family_id":"qf-mod34-statmodel-qualification","seed":202611034008,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,11],[1,11],[9,11]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/11, 1/11, 9/11], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/11, 1/11, 9/11], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/11, 1/11, 9/11], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0009","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbb7d4ef3e75a7a6e8b1ff9c93a7c3f1848500fcabac12d5d9d5bfcd79f0b25c","family_id":"qf-mod34-statmodel-qualification","seed":202611034009,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,12],[1,12],[10,12]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/12, 1/12, 10/12], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/12, 1/12, 5/6], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/12, 1/12, 5/6], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0010","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701d9f50754c7cedaf6919f2b9d9534dbc0519d5bcf1a002f78fd6a5a181b71c","family_id":"qf-mod34-statmodel-qualification","seed":202611034010,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,13],[1,13],[11,13]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/13, 1/13, 11/13], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/13, 1/13, 11/13], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/13, 1/13, 11/13], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0011","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd24237384b513589c1e9734bcc78bc6beab1f6c68e53207b89d84c4199fe54f","family_id":"qf-mod34-statmodel-qualification","seed":202611034011,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,14],[1,14],[12,14]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/14, 1/14, 12/14], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/14, 1/14, 6/7], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/14, 1/14, 6/7], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0012","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15dabc7ed766b9ada7d589f2d9ee05e3468fc5cd65d2a52499f106bd5be9a792","family_id":"qf-mod34-statmodel-qualification","seed":202611034012,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,15],[1,15],[13,15]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/15, 1/15, 13/15], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/15, 1/15, 13/15], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/15, 1/15, 13/15], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0013","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d27c4e66d67a112dd913d9f7fcd8a94c664800d553fb9b9c7801c60590a06f6f","family_id":"qf-mod34-statmodel-qualification","seed":202611034013,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,16],[1,16],[14,16]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/16, 1/16, 14/16], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/16, 1/16, 7/8], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/16, 1/16, 7/8], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0014","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8dd9914e15185e9df1dbe9504e2e0205cddc1d3c8d345c305a035042a36dd11","family_id":"qf-mod34-statmodel-qualification","seed":202611034014,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,17],[1,17],[15,17]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/17, 1/17, 15/17], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/17, 1/17, 15/17], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0015","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a97cbbe61978e826b4f1db7b9a27349d26c716cc1dbdfd1a00e2c44c6413242","family_id":"qf-mod34-statmodel-qualification","seed":202611034015,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,18],[1,18],[16,18]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/18, 1/18, 16/18], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/18, 1/18, 8/9], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/18, 1/18, 8/9], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0016","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ebc5dddac13dbaaf56b60eff60307e4aa52902298e78628d4040516d2ec5fae","family_id":"qf-mod34-statmodel-qualification","seed":202611034016,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/19, 1/19, 17/19], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/19, 1/19, 17/19], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0017","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f41cac55e8d747357e0da8521bee5365bfe4f818b43c60ef95d9a0599361e86c","family_id":"qf-mod34-statmodel-qualification","seed":202611034017,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,20],[1,20],[18,20]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/20, 1/20, 18/20], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/20, 1/20, 9/10], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/20, 1/20, 9/10], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0018","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54f1591f4db3aac805939a2d8a74328995b4336e3528ec56b8146762920c1dc5","family_id":"qf-mod34-statmodel-qualification","seed":202611034018,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,21],[1,21],[19,21]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/21, 1/21, 19/21], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/21, 1/21, 19/21], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/21, 1/21, 19/21], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0019","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b52dab0abeb3dcec4d0d4c20d3cca2947d0d5e09ba22dc9d869943bb58a248d","family_id":"qf-mod34-statmodel-qualification","seed":202611034019,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,22],[1,22],[20,22]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/22, 1/22, 20/22], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/22, 1/22, 10/11], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/22, 1/22, 10/11], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0020","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11466c2a019a37ebc16c20b90ff74e8afc53299cfe07f84b283a02ed0c353518","family_id":"qf-mod34-statmodel-qualification","seed":202611034020,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,23],[1,23],[21,23]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/23, 1/23, 21/23], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/23, 1/23, 21/23], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/23, 1/23, 21/23], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0021","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32c68db105460ad4170a73bc3fa36825e156c70c4fb12cda7e74464845e9b726","family_id":"qf-mod34-statmodel-qualification","seed":202611034021,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,24],[1,24],[22,24]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/24, 1/24, 22/24], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/24, 1/24, 11/12], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/24, 1/24, 11/12], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0022","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48522098945e9cc602fe12c6efb96fcf15033d7a66313f5d3935a68b7c49d327","family_id":"qf-mod34-statmodel-qualification","seed":202611034022,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,25],[1,25],[23,25]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/25, 1/25, 23/25], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/25, 1/25, 23/25], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/25, 1/25, 23/25], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0023","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4d5705f54c03802cbff3a5f3dccd39e22b1419cdeb1605fc5ae248ee56911a7","family_id":"qf-mod34-statmodel-qualification","seed":202611034023,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,26],[1,26],[24,26]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/26, 1/26, 24/26], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/26, 1/26, 12/13], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/26, 1/26, 12/13], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0024","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"447fdc1fb9af5977a4fa1c37ccbc22eb3edcd14b40ab13c750485ade3ad7e7b9","family_id":"qf-mod34-statmodel-qualification","seed":202611034024,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,27],[1,27],[25,27]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/27, 1/27, 25/27], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/27, 1/27, 25/27], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/27, 1/27, 25/27], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0025","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"516c4d3802fc269a1fbfd8504bb4806622e1a97a9b71580e031ac9fbb86be0b6","family_id":"qf-mod34-statmodel-qualification","seed":202611034025,"semantic_template":"qf-mod34-statmodel-qualification/{\"laws\":[[[1,28],[1,28],[26,28]],[[1,2],[1,3],[1,6]]],\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"Every Parameter Row Must Qualify on the Same Sample Space","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/28, 1/28, 26/28], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. Give (parameter_count,outcome_count,law_rows,qualified?,common_sample_space). Each row is [parameter,probability_entries,row_sum,row_qualified?] in original parameter order.","answer":{"kind":"expression","value":"(2, 3, [[0, [1/28, 1/28, 13/14], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)","explanation":"Audit every exact row sum and nonnegativity before conjoining qualification. Invalid candidate rows remain visible with their flags; calling an array a law table does not make it normalized. The exact result, retaining the requested order, is (2, 3, [[0, [1/28, 1/28, 13/14], 1, Yes], [1, [1/2, 1/3, 1/6], 1, Yes]], Yes, common_sample_space)."},"skills":["Every Parameter Row Must Qualify on the Same Sample Space"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0026","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02b38354cd6ac6d8d86e72596009cdc1733f44143414c6aee8b8e01ef1f58ef1","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034026,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,4],[1,4],[2,4]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/4, 1/4, 2/4], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 0, 1/4, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 0, 1/4, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0027","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72f44b987843a7f63ea3b289076b03b412edb315470d98700d519a6af76f61c9","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034027,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,5],[1,5],[3,5]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/5, 1/5, 3/5], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 1, 1/3, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 1, 1/3, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0028","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29da0479488346e658722fd29a7d4852f8e89dc62e2abfbc021c2ad63688e597","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034028,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,6],[1,6],[4,6]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/6, 1/6, 4/6], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 2, 2/3, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 2, 2/3, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0029","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32b7e7a39128b89122fb8886b8ec5ec7eee216e4d31e3568bdadf40456986a7f","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034029,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,7],[1,7],[5,7]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/7, 1/7, 5/7], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 0, 1/2, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 0, 1/2, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0030","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48f5c65cbff8e0392c3fe0d308a37f84af1d76fd5939da60c93ee9c659b22ffd","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034030,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,8],[1,8],[6,8]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/8, 1/8, 6/8], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 1, 1/8, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 1, 1/8, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0031","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d9f947598ede3193a2cf9e010ea8eeb0574d3c00658896df68d39a8f1e3816","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034031,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,9],[1,9],[7,9]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 2, 1/6, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 2, 1/6, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0032","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1167d39bc122d5827164205a39fe9ddd7ce37966be9ba6d65611e93629a48074","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034032,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,10],[1,10],[8,10]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/10, 1/10, 8/10], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 0, 1/10, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 0, 1/10, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0033","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20ab6761de561c12c8335e89ad12416fd3e85e5854bb64544a2af6b23f3ded71","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034033,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,11],[1,11],[9,11]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/11, 1/11, 9/11], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 1, 1/3, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 1, 1/3, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0034","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e65700ee859a57bca8d10ad66a1c2bb1467db87e94869c5ad78acd8e861f9a0","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034034,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,12],[1,12],[10,12]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/12, 1/12, 10/12], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 2, 5/6, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 2, 5/6, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0035","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d72a8192a9f4b47f5d55b09998728188ed15fa78ac3183348f2c76e871429a39","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034035,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,13],[1,13],[11,13]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/13, 1/13, 11/13], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 0, 1/2, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 0, 1/2, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0036","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c5d07509e2769adc0c7fd406fb98495f6dcd0d3c61c1b2ac5b45e6dae132f82","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034036,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,14],[1,14],[12,14]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/14, 1/14, 12/14], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 1, 1/14, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 1, 1/14, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0037","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"736d6c0dd2d14cb05b2c81864b37946c414c614dd70705ef37821941528d6188","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034037,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,15],[1,15],[13,15]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/15, 1/15, 13/15], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 2, 1/6, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 2, 1/6, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0038","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f9c232fedcbdb050e311647128f096c496ebdb7f93b962127a23432239ae5d2","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034038,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,16],[1,16],[14,16]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/16, 1/16, 14/16], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 0, 1/16, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 0, 1/16, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0039","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7736fd5040d08b813c171479160e75c31bf6503c53cd187a8b1f39d83f873689","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034039,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,17],[1,17],[15,17]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 1, 1/3, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 1, 1/3, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0040","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bbeceaaad146aa1303d437bcae86fc15e1c6aed76a1a47db37f216b78f6065f","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034040,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,18],[1,18],[16,18]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/18, 1/18, 16/18], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 2, 8/9, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 2, 8/9, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0041","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a058342141d258f2f443cf5395d12ec155096e06e8b5230e2c57eb505ce98581","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034041,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 0, 1/2, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 0, 1/2, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0042","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6115bccd3fd09592dbdcfa3c1b8108527d5cb2b105f9d2a1dec0cc5fe3ebc2c5","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034042,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,20],[1,20],[18,20]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/20, 1/20, 18/20], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 1, 1/20, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 1, 1/20, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0043","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62a4c52cba3159eabb5260fd0a116bcf409f5b49ef0237210733a790432cc3a6","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034043,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,21],[1,21],[19,21]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/21, 1/21, 19/21], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 2, 1/6, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 2, 1/6, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0044","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd98f81c92053e973984f26ac13f6f43ce8f926da0c3c7660983a478cf69ba08","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034044,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,22],[1,22],[20,22]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/22, 1/22, 20/22], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 0, 1/22, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 0, 1/22, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0045","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2799994efed2ac09b6b5fd0b354834927ccb7d5788a106996b7d21766d9c1a63","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034045,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,23],[1,23],[21,23]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/23, 1/23, 21/23], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 1, 1/3, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 1, 1/3, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0046","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ee086917ba24edf29112bc6b10c0256d7992141f3c6854d8f4408365440ab45","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034046,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,24],[1,24],[22,24]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/24, 1/24, 22/24], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 2, 11/12, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 2, 11/12, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0047","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8587780e8d289da7ae437edc1f827d0879f09c8c63571d882f10b8f25efd05c4","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034047,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,25],[1,25],[23,25]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/25, 1/25, 23/25], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 0, 1/2, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 0, 1/2, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0048","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0270a0b9c8c34a41dcdb780612a68c7444c884773608b1e6253567c6e93c7ea","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034048,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,26],[1,26],[24,26]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/26, 1/26, 24/26], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 1, 1/26, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 1, 1/26, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0049","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57066d7ad8f13cad5adba2ef2ef6ce69882e0a519fe96b984be2f4bd70d57cb2","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034049,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,27],[1,27],[25,27]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/27, 1/27, 25/27], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(1, 2, 1/6, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (1, 2, 1/6, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0050","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0872bcf7cfefeb3da18e762782943dea09823532c99342d98801ad72f6d1c2bb","family_id":"qf-mod34-statmodel-law-lookup","seed":202611034050,"semantic_template":"qf-mod34-statmodel-law-lookup/{\"laws\":[[[1,28],[1,28],[26,28]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Read One Parameter-Outcome Cell without Confusing Their Roles","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/28, 1/28, 26/28], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,probability,model_qualified?).","answer":{"kind":"expression","value":"(0, 0, 1/28, Yes)","explanation":"Select the row by parameter and column by observed outcome, then retain whole-table qualification separately. A raw cell from an invalid candidate table does not repair the model or become a posterior. The exact result, retaining the requested order, is (0, 0, 1/28, Yes)."},"skills":["Read One Parameter-Outcome Cell without Confusing Their Roles"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0051","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c2c19407cb7873fe9a91b73ce9de7c23969aa222e56d0e785a86ba965758544","family_id":"qf-mod34-statmodel-likelihood","seed":202611034051,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,4],[1,4],[2,4]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/4, 1/4, 2/4], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/4], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/4], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0052","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d742ed5982c36fc04bbdbc963ce9b627ca1ce5f595df3a2e0261d55f4e2c0d1","family_id":"qf-mod34-statmodel-likelihood","seed":202611034052,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,5],[1,5],[3,5]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/5, 1/5, 3/5], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/5], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/5], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0053","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"711ad7cdfaa04ed5b3bfd334c49768e80366cb711d7356c74f095dbd8e4ced80","family_id":"qf-mod34-statmodel-likelihood","seed":202611034053,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,6],[1,6],[4,6]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/6, 1/6, 4/6], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 2/3], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 2/3], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0054","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34f97702a36f6dd0eacaa05285b69d17a1c9a6291075bbc1f5856b8aeb81e964","family_id":"qf-mod34-statmodel-likelihood","seed":202611034054,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,7],[1,7],[5,7]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/7, 1/7, 5/7], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/7], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/7], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0055","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71f6ad546a615303de8e1233da59412d9b913eb1940eacc1517e44f8d5a73936","family_id":"qf-mod34-statmodel-likelihood","seed":202611034055,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,8],[1,8],[6,8]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/8, 1/8, 6/8], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/8], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/8], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0056","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83dcbdefb4a4dd28fd783ddeade49af3c33b990ab035645f5303bc3beae65b29","family_id":"qf-mod34-statmodel-likelihood","seed":202611034056,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,9],[1,9],[7,9]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 7/9], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 7/9], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0057","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cecefcd3859a5c2d5b0270d1ed37570596a2f76736fec2a398167fd33b9b389f","family_id":"qf-mod34-statmodel-likelihood","seed":202611034057,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,10],[1,10],[8,10]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/10, 1/10, 8/10], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/10], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/10], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0058","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fea166a6d7eee8b1b068db13c659986e36d1e356ce51cdcd3600d489a1de6c11","family_id":"qf-mod34-statmodel-likelihood","seed":202611034058,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,11],[1,11],[9,11]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/11, 1/11, 9/11], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/11], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/11], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0059","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"753576f3a92dc862e1f37ea9d33e24b9ceaa088ed3fb546d91bf0579af3b57a0","family_id":"qf-mod34-statmodel-likelihood","seed":202611034059,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,12],[1,12],[10,12]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/12, 1/12, 10/12], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 5/6], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 5/6], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0060","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"246fee9074aef5a230ce9c551e4d3e852a78491d45a4219e6ca0cce1407ea14b","family_id":"qf-mod34-statmodel-likelihood","seed":202611034060,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,13],[1,13],[11,13]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/13, 1/13, 11/13], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/13], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/13], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0061","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e44fe697a856bd9336347fb7abdf1796252f58145e3d75d8c170f006ae2896a9","family_id":"qf-mod34-statmodel-likelihood","seed":202611034061,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,14],[1,14],[12,14]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/14, 1/14, 12/14], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/14], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/14], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0062","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d964a81c8188751dd78b0f36533d9f8927f3cac38a78de9eca32a9ca17952e6","family_id":"qf-mod34-statmodel-likelihood","seed":202611034062,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,15],[1,15],[13,15]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/15, 1/15, 13/15], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 13/15], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 13/15], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0063","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"760963d44169347f99bc5d56cc4a272c173dd87364679c996288235aee1f9319","family_id":"qf-mod34-statmodel-likelihood","seed":202611034063,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,16],[1,16],[14,16]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/16, 1/16, 14/16], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/16], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/16], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0064","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e740534801b0f42a34e32243476d6e272dd79b474707e65c0a75fafb5ef03b9d","family_id":"qf-mod34-statmodel-likelihood","seed":202611034064,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,17],[1,17],[15,17]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/17], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/17], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0065","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1d081252be6e2a925748ab9f85b7b1eaffe3af05435e20b5a9a4f5a5777edc6","family_id":"qf-mod34-statmodel-likelihood","seed":202611034065,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,18],[1,18],[16,18]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/18, 1/18, 16/18], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 8/9], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 8/9], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0066","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d30ae75bed4fb9a8767ffc185e82f084f3a77d4f7f70115a84012ca31ef83542","family_id":"qf-mod34-statmodel-likelihood","seed":202611034066,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/19], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/19], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0067","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f9a3766b071b833caefa88a225ef179f7168aea8652c04d32bd59bcdffe98d","family_id":"qf-mod34-statmodel-likelihood","seed":202611034067,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,20],[1,20],[18,20]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/20, 1/20, 18/20], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/20], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/20], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0068","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1bba7620f09e442537b102f6041f3db091c51e0cfc40e1aa11c33dac4347741","family_id":"qf-mod34-statmodel-likelihood","seed":202611034068,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,21],[1,21],[19,21]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/21, 1/21, 19/21], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 19/21], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 19/21], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0069","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96243a8fd227e096aa259fe1cf06dab1d11f06f260f72dc6c5beff2f4dbeb8cb","family_id":"qf-mod34-statmodel-likelihood","seed":202611034069,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,22],[1,22],[20,22]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/22, 1/22, 20/22], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/22], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/22], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0070","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a048f324e863ddcb9e1dc69ef1f775312d996705d85e83879f7bf62eca9fc1eb","family_id":"qf-mod34-statmodel-likelihood","seed":202611034070,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,23],[1,23],[21,23]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/23, 1/23, 21/23], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/23], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/23], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0071","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02746bac15685ab71488db8872ff59f9e0dfc88797fcc96fadf386f1d6048fa4","family_id":"qf-mod34-statmodel-likelihood","seed":202611034071,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,24],[1,24],[22,24]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/24, 1/24, 22/24], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 11/12], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 11/12], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0072","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"663bd56be1356d376a5e6787f84c615202bbff12131ed1837140b3c15229a7ea","family_id":"qf-mod34-statmodel-likelihood","seed":202611034072,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,25],[1,25],[23,25]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/25, 1/25, 23/25], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/25], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/25], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0073","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c16ec6698a8734ef4c6363b4c0af24f2ef322ca854703247d7c8179aaa5d946e","family_id":"qf-mod34-statmodel-likelihood","seed":202611034073,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,26],[1,26],[24,26]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/26, 1/26, 24/26], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=1. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(1, [[0, 1/26], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (1, [[0, 1/26], [1, 1/3]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0074","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ca51cacc44ea07e12a796386833b6266f03c430505796b7179eb4a4e927ea8a","family_id":"qf-mod34-statmodel-likelihood","seed":202611034074,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,27],[1,27],[25,27]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/27, 1/27, 25/27], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=2. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(2, [[0, 25/27], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (2, [[0, 25/27], [1, 1/6]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0075","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f9e0b1db6a64e6608a42c1bb57db14bb9199423c4e2bad21a545cbe2b7e3494e","family_id":"qf-mod34-statmodel-likelihood","seed":202611034075,"semantic_template":"qf-mod34-statmodel-likelihood/{\"laws\":[[[1,28],[1,28],[26,28]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter_count\":2}"},"fields":{"title":"A Likelihood Varies the Parameter for One Fixed Observation","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/28, 1/28, 26/28], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. In the qualified model fix observed outcome=0. Give (outcome,likelihood_rows,likelihood_as_function_of_parameter_not_posterior), with rows [parameter,P_parameter(outcome)] in parameter order.","answer":{"kind":"expression","value":"(0, [[0, 1/28], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)","explanation":"Read that same outcome column across all parameter rows. These values need not sum to one over parameters and are not posterior probabilities; no parameter prior was supplied. The exact result, retaining the requested order, is (0, [[0, 1/28], [1, 1/2]], likelihood_as_function_of_parameter_not_posterior)."},"skills":["A Likelihood Varies the Parameter for One Fixed Observation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0076","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a147c748735854ecfa95cffffebc9114649aef9695c0c150ec6927947d4316b6","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034076,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,4],[1,4],[2,4]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/4, 1/4, 2/4], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 0, parameter_index, observed_outcome, 1/4, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 0, parameter_index, observed_outcome, 1/4, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0077","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee8a301c6c988d9008b35c2e20c8313e0b5ff38ea77ec00cef87ee1022e43b62","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034077,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,5],[1,5],[3,5]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/5, 1/5, 3/5], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0078","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5c9317f3fa355df041ec6bef37cbd1e9c286077e9a386e79df5267df2b049c6","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034078,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,6],[1,6],[4,6]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/6, 1/6, 4/6], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 2, parameter_index, observed_outcome, 2/3, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 2, parameter_index, observed_outcome, 2/3, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0079","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4f0b7aec681945864a3aa7bbaf866fb5723ea80ee73c7359f055d2c6898c1b","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034079,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,7],[1,7],[5,7]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/7, 1/7, 5/7], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0080","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aadf24a8e0808c83c08bb37ec0c030223bb614819660008d85c76f1732eadcab","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034080,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,8],[1,8],[6,8]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/8, 1/8, 6/8], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 1, parameter_index, observed_outcome, 1/8, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 1, parameter_index, observed_outcome, 1/8, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0081","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4196607cd3d66c9bd981718bf1a9a770095c0cb4ec740e9308932ee72e9d4e5d","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034081,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,9],[1,9],[7,9]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/9, 1/9, 7/9], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0082","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3e0a6f05c6722254292b848820be87ceb54e99418dca1faab6d48d160cd2044","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034082,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,10],[1,10],[8,10]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/10, 1/10, 8/10], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 0, parameter_index, observed_outcome, 1/10, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 0, parameter_index, observed_outcome, 1/10, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0083","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f6fee0d09e9b4b31eeb0c73830d18b7c4df8c238fc8ee0d7b61e9e6449acdd6","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034083,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,11],[1,11],[9,11]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/11, 1/11, 9/11], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0084","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17c1bf5ba1d1c21e5c7987677f857abfd43d808154a46bcc021e75a01fe56c9d","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034084,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,12],[1,12],[10,12]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/12, 1/12, 10/12], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 2, parameter_index, observed_outcome, 5/6, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 2, parameter_index, observed_outcome, 5/6, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0085","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"319e3debd46d6a627c8c6939147a0fa6afb384eb5f3c9b4573fdea0dd9194c52","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034085,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,13],[1,13],[11,13]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/13, 1/13, 11/13], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0086","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e859de2a37d888fa95925d45b7124b4674bd57d400bd6a9107961fc5fb6b64e1","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034086,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,14],[1,14],[12,14]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/14, 1/14, 12/14], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 1, parameter_index, observed_outcome, 1/14, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 1, parameter_index, observed_outcome, 1/14, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0087","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5bc808acd9c1ef99aada810d6a401dd50ce7453ca081fb5577a4be047ab52f6","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034087,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,15],[1,15],[13,15]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/15, 1/15, 13/15], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0088","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae2146cb3baa03b348bd4e64c8cfea4cca84aaa1a5e8f23f8cfee26006aeea22","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034088,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,16],[1,16],[14,16]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/16, 1/16, 14/16], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 0, parameter_index, observed_outcome, 1/16, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 0, parameter_index, observed_outcome, 1/16, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0089","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9052a726176ac0fca16e9869c20effb04c0f51311b1c3adca80a371f033290f6","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034089,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,17],[1,17],[15,17]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/17, 1/17, 15/17], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0090","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e11b2cdf3a446f0c86231dac2bc3a879faece98a7ccd36414dcd33bbe74f92cc","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034090,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,18],[1,18],[16,18]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/18, 1/18, 16/18], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 2, parameter_index, observed_outcome, 8/9, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 2, parameter_index, observed_outcome, 8/9, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0091","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd983da8d546de485ac633f722a90283f8ddc5ade7c9e8fbe9f10e869a9c322f","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034091,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,19],[1,19],[17,19]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/19, 1/19, 17/19], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0092","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fea22ebe6528d6a3353db20a60ac1657f780180a7d3f2a551cf1cdc98c42763e","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034092,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,20],[1,20],[18,20]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/20, 1/20, 18/20], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 1, parameter_index, observed_outcome, 1/20, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 1, parameter_index, observed_outcome, 1/20, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0093","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb69bb317e267b003280b8427acadceb7f344b469c7c5047e50cbae06b58f816","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034093,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,21],[1,21],[19,21]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/21, 1/21, 19/21], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0094","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4126158ae6ae1d5f3ba8b20e3f6894e40a870de8438d7a0069ccaa5b358e24de","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034094,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,22],[1,22],[20,22]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/22, 1/22, 20/22], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 0, parameter_index, observed_outcome, 1/22, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 0, parameter_index, observed_outcome, 1/22, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0095","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa22987172ce9307735f5069a92f028d274a5ecf31a4dcdc047a59b82cb073bb","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034095,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,23],[1,23],[21,23]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/23, 1/23, 21/23], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 1, parameter_index, observed_outcome, 1/3, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0096","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55071f6d9e6fb192c8e8d31c327c7050e299da63c71e3e22acc5da6ef59e866c","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034096,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,24],[1,24],[22,24]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/24, 1/24, 22/24], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 2, parameter_index, observed_outcome, 11/12, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 2, parameter_index, observed_outcome, 11/12, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0097","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e543da32162c57afaccd2b443a97129d0c2c5c1db114190e77d923fafcf48f","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034097,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,25],[1,25],[23,25]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/25, 1/25, 23/25], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 0, parameter_index, observed_outcome, 1/2, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0098","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a56f6eaf49fe78b1c04af74dbc57829efcd64fb95bdbb20df9fcea6ab1c99ce1","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034098,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,26],[1,26],[24,26]],[[1,2],[1,3],[1,6]]],\"outcome\":1,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/26, 1/26, 24/26], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=1. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 1, parameter_index, observed_outcome, 1/26, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 1, parameter_index, observed_outcome, 1/26, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0099","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23143092a2b8a45df0f00856fd0df2d8fe70445c969f33ba96d783b3b341b36b","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034099,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,27],[1,27],[25,27]],[[1,2],[1,3],[1,6]]],\"outcome\":2,\"outcome_count\":3,\"parameter\":1,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/27, 1/27, 25/27], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=1, outcome=2. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (1, 2, parameter_index, observed_outcome, 1/6, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261103-r04-parametric-statistical-model-0100","batch_id":"batch-20261014-advance-20261103-r04-parametric-statistical-model-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b0573ecb8f7922698693243de819cdd370a5557951bea18f0f0b57023e4500f","family_id":"qf-mod34-statmodel-parameter-observation-boundary","seed":202611034100,"semantic_template":"qf-mod34-statmodel-parameter-observation-boundary/{\"laws\":[[[1,28],[1,28],[26,28]],[[1,2],[1,3],[1,6]]],\"outcome\":0,\"outcome_count\":3,\"parameter\":0,\"parameter_count\":2}"},"fields":{"title":"Equal Numeric Indices Do Not Give Parameters and Observations the Same Role","stem":"Use parameter indices=Fin(2)={0,…,1} and one shared outcome space=Fin(3)={0,…,2}, with the complete parameter-by-outcome candidate law table=[[1/28, 1/28, 26/28], [1/2, 1/3, 1/6]]. Every row must be nonnegative and sum exactly to 1 before the table qualifies as a statistical model. Keep parameter and observation roles distinct even when their numeric indices agree. A likelihood fixes the observed outcome and varies the parameter; no prior, posterior, identifiability or inferential optimality is implied. Rational computations are exact. parameter=0, outcome=0. Give (parameter,outcome,parameter_index,observed_outcome,law_value,same_role?,parameter_not_observation_and_likelihood_not_posterior).","answer":{"kind":"expression","value":"(0, 0, parameter_index, observed_outcome, 1/28, No, parameter_not_observation_and_likelihood_not_posterior)","explanation":"Read the cell under the stated parameter. The same_role No is about semantic roles, even when the numeric labels coincide. It does not classify a likelihood as a posterior or an observed outcome as an unknown parameter. The exact result, retaining the requested order, is (0, 0, parameter_index, observed_outcome, 1/28, No, parameter_not_observation_and_likelihood_not_posterior)."},"skills":["Equal Numeric Indices Do Not Give Parameters and Observations the Same Role"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}