{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0001","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"values\":[0,1,3,99],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], 是, 23/10)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], 是, 23/10)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094001,"parameters":{"masses":[[1,10],[2,10],[7,10],[0,1]],"values":[0,1,3,99],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0002","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"values\":[1,2,4,100],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], 是, 37/11)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], 是, 37/11)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094002,"parameters":{"masses":[[1,11],[2,11],[8,11],[0,1]],"values":[1,2,4,100],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0003","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"values\":[2,3,5,101],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], 是, 53/12)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], 是, 53/12)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094003,"parameters":{"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0004","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"values\":[3,4,6,102],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], 是, 71/13)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], 是, 71/13)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094004,"parameters":{"masses":[[1,13],[2,13],[10,13],[0,1]],"values":[3,4,6,102],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0005","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"values\":[4,5,7,103],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], 是, 13/2)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], 是, 13/2)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094005,"parameters":{"masses":[[1,14],[2,14],[11,14],[0,1]],"values":[4,5,7,103],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0006","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"values\":[5,6,8,104],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], 是, 113/15)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], 是, 113/15)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094006,"parameters":{"masses":[[1,15],[2,15],[12,15],[0,1]],"values":[5,6,8,104],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0007","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"values\":[6,7,9,105],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], 是, 137/16)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], 是, 137/16)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094007,"parameters":{"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0008","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"values\":[7,8,10,106],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], 是, 163/17)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], 是, 163/17)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094008,"parameters":{"masses":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0009","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"values\":[8,9,11,107],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], 是, 191/18)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], 是, 191/18)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094009,"parameters":{"masses":[[1,18],[2,18],[15,18],[0,1]],"values":[8,9,11,107],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0010","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"values\":[9,10,12,108],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], 是, 221/19)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], 是, 221/19)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094010,"parameters":{"masses":[[1,19],[2,19],[16,19],[0,1]],"values":[9,10,12,108],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0011","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"values\":[10,11,13,109],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], 是, 253/20)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], 是, 253/20)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094011,"parameters":{"masses":[[1,20],[2,20],[17,20],[0,1]],"values":[10,11,13,109],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0012","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], 是, 41/3)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], 是, 41/3)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094012,"parameters":{"masses":[[1,21],[2,21],[18,21],[0,1]],"values":[11,12,14,110],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0013","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"values\":[12,13,15,111],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], 是, 323/22)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], 是, 323/22)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094013,"parameters":{"masses":[[1,22],[2,22],[19,22],[0,1]],"values":[12,13,15,111],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0014","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"values\":[13,14,16,112],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], 是, 361/23)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], 是, 361/23)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094014,"parameters":{"masses":[[1,23],[2,23],[20,23],[0,1]],"values":[13,14,16,112],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0015","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"values\":[14,15,17,113],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], 是, 401/24)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], 是, 401/24)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094015,"parameters":{"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0016","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"values\":[15,16,18,114],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], 是, 443/25)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], 是, 443/25)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094016,"parameters":{"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0017","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"values\":[16,17,19,115],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], 是, 487/26)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], 是, 487/26)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094017,"parameters":{"masses":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0018","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"values\":[17,18,20,116],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], 是, 533/27)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], 是, 533/27)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094018,"parameters":{"masses":[[1,27],[2,27],[24,27],[0,1]],"values":[17,18,20,116],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0019","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"values\":[18,19,21,117],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], 是, 83/4)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], 是, 83/4)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094019,"parameters":{"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0020","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"values\":[19,20,22,118],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], 是, 631/29)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], 是, 631/29)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094020,"parameters":{"masses":[[1,29],[2,29],[26,29],[0,1]],"values":[19,20,22,118],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0021","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"values\":[20,21,23,119],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], 是, 683/30)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], 是, 683/30)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094021,"parameters":{"masses":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0022","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"values\":[21,22,24,120],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], 是, 737/31)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], 是, 737/31)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094022,"parameters":{"masses":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0023","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], 是, 793/32)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], 是, 793/32)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094023,"parameters":{"masses":[[1,32],[2,32],[29,32],[0,1]],"values":[22,23,25,121],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0024","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"values\":[23,24,26,122],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], 是, 851/33)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], 是, 851/33)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094024,"parameters":{"masses":[[1,33],[2,33],[30,33],[0,1]],"values":[23,24,26,122],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0025","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-partition-audit；資料={\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"values\":[24,25,27,123],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], 是, 911/34)","explanation_zh":"Wave6第三層核對=([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], 是, 911/34)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-partition-audit","seed":202611094025,"parameters":{"masses":[[1,34],[2,34],[31,34],[0,1]],"values":[24,25,27,123],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-partition-audit","semantic_template":"qf-mod40-totalexp-partition-audit/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0026","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"values\":[0,1,3,99],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094026,"parameters":{"masses":[[1,10],[2,10],[7,10],[0,1]],"values":[0,1,3,99],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0027","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"values\":[1,2,4,100],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094027,"parameters":{"masses":[[1,11],[2,11],[8,11],[0,1]],"values":[1,2,4,100],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0028","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"values\":[2,3,5,101],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094028,"parameters":{"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0029","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"values\":[3,4,6,102],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094029,"parameters":{"masses":[[1,13],[2,13],[10,13],[0,1]],"values":[3,4,6,102],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0030","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"values\":[4,5,7,103],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094030,"parameters":{"masses":[[1,14],[2,14],[11,14],[0,1]],"values":[4,5,7,103],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0031","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"values\":[5,6,8,104],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094031,"parameters":{"masses":[[1,15],[2,15],[12,15],[0,1]],"values":[5,6,8,104],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0032","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"values\":[6,7,9,105],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094032,"parameters":{"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0033","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"values\":[7,8,10,106],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094033,"parameters":{"masses":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0034","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"values\":[8,9,11,107],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094034,"parameters":{"masses":[[1,18],[2,18],[15,18],[0,1]],"values":[8,9,11,107],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0035","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"values\":[9,10,12,108],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094035,"parameters":{"masses":[[1,19],[2,19],[16,19],[0,1]],"values":[9,10,12,108],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0036","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"values\":[10,11,13,109],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094036,"parameters":{"masses":[[1,20],[2,20],[17,20],[0,1]],"values":[10,11,13,109],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0037","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094037,"parameters":{"masses":[[1,21],[2,21],[18,21],[0,1]],"values":[11,12,14,110],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0038","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"values\":[12,13,15,111],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094038,"parameters":{"masses":[[1,22],[2,22],[19,22],[0,1]],"values":[12,13,15,111],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0039","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"values\":[13,14,16,112],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094039,"parameters":{"masses":[[1,23],[2,23],[20,23],[0,1]],"values":[13,14,16,112],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0040","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"values\":[14,15,17,113],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094040,"parameters":{"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0041","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"values\":[15,16,18,114],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094041,"parameters":{"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0042","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"values\":[16,17,19,115],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094042,"parameters":{"masses":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0043","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"values\":[17,18,20,116],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094043,"parameters":{"masses":[[1,27],[2,27],[24,27],[0,1]],"values":[17,18,20,116],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0044","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"values\":[18,19,21,117],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094044,"parameters":{"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0045","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"values\":[19,20,22,118],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094045,"parameters":{"masses":[[1,29],[2,29],[26,29],[0,1]],"values":[19,20,22,118],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0046","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"values\":[20,21,23,119],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094046,"parameters":{"masses":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0047","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"values\":[21,22,24,120],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094047,"parameters":{"masses":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0048","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094048,"parameters":{"masses":[[1,32],[2,32],[29,32],[0,1]],"values":[22,23,25,121],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0049","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"values\":[23,24,26,122],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, 是)","explanation_zh":"Wave6第三層核對=([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094049,"parameters":{"masses":[[1,33],[2,33],[30,33],[0,1]],"values":[23,24,26,122],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0050","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-positive-cells-formula；資料={\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"values\":[24,25,27,123],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, 是)","explanation_zh":"Wave6第三層核對=([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, 是)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-positive-cells-formula","seed":202611094050,"parameters":{"masses":[[1,34],[2,34],[31,34],[0,1]],"values":[24,25,27,123],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-positive-cells-formula","semantic_template":"qf-mod40-totalexp-positive-cells-formula/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0051","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"values\":[0,1,3,99],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([2/3, 2/3, 3, 3], 23/10, 23/10, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([2/3, 2/3, 3, 3], 23/10, 23/10, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094051,"parameters":{"masses":[[1,10],[2,10],[7,10],[0,1]],"values":[0,1,3,99],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[0,1,3,99]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0052","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"values\":[1,2,4,100],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([5/3, 5/3, 4, 4], 37/11, 37/11, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([5/3, 5/3, 4, 4], 37/11, 37/11, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094052,"parameters":{"masses":[[1,11],[2,11],[8,11],[0,1]],"values":[1,2,4,100],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[1,2,4,100]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0053","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"values\":[2,3,5,101],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([8/3, 8/3, 5, 5], 53/12, 53/12, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([8/3, 8/3, 5, 5], 53/12, 53/12, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094053,"parameters":{"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[2,3,5,101]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0054","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"values\":[3,4,6,102],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([11/3, 11/3, 6, 6], 71/13, 71/13, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([11/3, 11/3, 6, 6], 71/13, 71/13, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094054,"parameters":{"masses":[[1,13],[2,13],[10,13],[0,1]],"values":[3,4,6,102],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[3,4,6,102]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0055","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"values\":[4,5,7,103],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([14/3, 14/3, 7, 7], 13/2, 13/2, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([14/3, 14/3, 7, 7], 13/2, 13/2, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094055,"parameters":{"masses":[[1,14],[2,14],[11,14],[0,1]],"values":[4,5,7,103],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[4,5,7,103]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0056","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"values\":[5,6,8,104],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([17/3, 17/3, 8, 8], 113/15, 113/15, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([17/3, 17/3, 8, 8], 113/15, 113/15, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094056,"parameters":{"masses":[[1,15],[2,15],[12,15],[0,1]],"values":[5,6,8,104],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[5,6,8,104]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0057","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"values\":[6,7,9,105],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([20/3, 20/3, 9, 9], 137/16, 137/16, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([20/3, 20/3, 9, 9], 137/16, 137/16, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094057,"parameters":{"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[6,7,9,105]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0058","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"values\":[7,8,10,106],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([23/3, 23/3, 10, 10], 163/17, 163/17, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([23/3, 23/3, 10, 10], 163/17, 163/17, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094058,"parameters":{"masses":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[7,8,10,106]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0059","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"values\":[8,9,11,107],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([26/3, 26/3, 11, 11], 191/18, 191/18, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([26/3, 26/3, 11, 11], 191/18, 191/18, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094059,"parameters":{"masses":[[1,18],[2,18],[15,18],[0,1]],"values":[8,9,11,107],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[8,9,11,107]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0060","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"values\":[9,10,12,108],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([29/3, 29/3, 12, 12], 221/19, 221/19, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([29/3, 29/3, 12, 12], 221/19, 221/19, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094060,"parameters":{"masses":[[1,19],[2,19],[16,19],[0,1]],"values":[9,10,12,108],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[9,10,12,108]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0061","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"values\":[10,11,13,109],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([32/3, 32/3, 13, 13], 253/20, 253/20, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([32/3, 32/3, 13, 13], 253/20, 253/20, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094061,"parameters":{"masses":[[1,20],[2,20],[17,20],[0,1]],"values":[10,11,13,109],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[10,11,13,109]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0062","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([35/3, 35/3, 14, 14], 41/3, 41/3, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([35/3, 35/3, 14, 14], 41/3, 41/3, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094062,"parameters":{"masses":[[1,21],[2,21],[18,21],[0,1]],"values":[11,12,14,110],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[11,12,14,110]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0063","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"values\":[12,13,15,111],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([38/3, 38/3, 15, 15], 323/22, 323/22, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([38/3, 38/3, 15, 15], 323/22, 323/22, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094063,"parameters":{"masses":[[1,22],[2,22],[19,22],[0,1]],"values":[12,13,15,111],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[12,13,15,111]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0064","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"values\":[13,14,16,112],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([41/3, 41/3, 16, 16], 361/23, 361/23, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([41/3, 41/3, 16, 16], 361/23, 361/23, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094064,"parameters":{"masses":[[1,23],[2,23],[20,23],[0,1]],"values":[13,14,16,112],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[13,14,16,112]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0065","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"values\":[14,15,17,113],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([44/3, 44/3, 17, 17], 401/24, 401/24, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([44/3, 44/3, 17, 17], 401/24, 401/24, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094065,"parameters":{"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[14,15,17,113]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0066","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"values\":[15,16,18,114],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([47/3, 47/3, 18, 18], 443/25, 443/25, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([47/3, 47/3, 18, 18], 443/25, 443/25, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094066,"parameters":{"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[15,16,18,114]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0067","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"values\":[16,17,19,115],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([50/3, 50/3, 19, 19], 487/26, 487/26, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([50/3, 50/3, 19, 19], 487/26, 487/26, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094067,"parameters":{"masses":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[16,17,19,115]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0068","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"values\":[17,18,20,116],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([53/3, 53/3, 20, 20], 533/27, 533/27, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([53/3, 53/3, 20, 20], 533/27, 533/27, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094068,"parameters":{"masses":[[1,27],[2,27],[24,27],[0,1]],"values":[17,18,20,116],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[17,18,20,116]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0069","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"values\":[18,19,21,117],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([56/3, 56/3, 21, 21], 83/4, 83/4, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([56/3, 56/3, 21, 21], 83/4, 83/4, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094069,"parameters":{"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[18,19,21,117]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0070","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"values\":[19,20,22,118],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([59/3, 59/3, 22, 22], 631/29, 631/29, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([59/3, 59/3, 22, 22], 631/29, 631/29, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094070,"parameters":{"masses":[[1,29],[2,29],[26,29],[0,1]],"values":[19,20,22,118],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[19,20,22,118]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0071","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"values\":[20,21,23,119],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([62/3, 62/3, 23, 23], 683/30, 683/30, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([62/3, 62/3, 23, 23], 683/30, 683/30, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094071,"parameters":{"masses":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[20,21,23,119]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0072","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"values\":[21,22,24,120],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([65/3, 65/3, 24, 24], 737/31, 737/31, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([65/3, 65/3, 24, 24], 737/31, 737/31, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094072,"parameters":{"masses":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[21,22,24,120]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0073","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([68/3, 68/3, 25, 25], 793/32, 793/32, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([68/3, 68/3, 25, 25], 793/32, 793/32, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094073,"parameters":{"masses":[[1,32],[2,32],[29,32],[0,1]],"values":[22,23,25,121],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[22,23,25,121]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0074","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"values\":[23,24,26,122],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([71/3, 71/3, 26, 26], 851/33, 851/33, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([71/3, 71/3, 26, 26], 851/33, 851/33, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094074,"parameters":{"masses":[[1,33],[2,33],[30,33],[0,1]],"values":[23,24,26,122],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[23,24,26,122]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0075","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-tower-identity；資料={\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"values\":[24,25,27,123],\"partition\":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([74/3, 74/3, 27, 27], 911/34, 911/34, 是, conditional_expectation_constant_on_partition_cells)","explanation_zh":"Wave6第三層核對=([74/3, 74/3, 27, 27], 911/34, 911/34, 是, conditional_expectation_constant_on_partition_cells)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-tower-identity","seed":202611094075,"parameters":{"masses":[[1,34],[2,34],[31,34],[0,1]],"values":[24,25,27,123],"partition":[[0,1],[2,3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-tower-identity","semantic_template":"qf-mod40-totalexp-tower-identity/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1],[2,3]],\"values\":[24,25,27,123]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0076","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"values\":[0,1,3,99],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094076,"parameters":{"masses":[[1,10],[2,10],[7,10],[0,1]],"values":[0,1,3,99],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,10],[2,10],[7,10],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[0,1,3,99]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0077","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"values\":[1,2,4,100],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094077,"parameters":{"masses":[[1,11],[2,11],[8,11],[0,1]],"values":[1,2,4,100],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,11],[2,11],[8,11],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[1,2,4,100]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0078","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"values\":[2,3,5,101],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094078,"parameters":{"masses":[[1,12],[2,12],[9,12],[0,1]],"values":[2,3,5,101],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,12],[2,12],[9,12],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[2,3,5,101]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0079","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"values\":[3,4,6,102],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094079,"parameters":{"masses":[[1,13],[2,13],[10,13],[0,1]],"values":[3,4,6,102],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,13],[2,13],[10,13],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[3,4,6,102]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0080","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"values\":[4,5,7,103],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094080,"parameters":{"masses":[[1,14],[2,14],[11,14],[0,1]],"values":[4,5,7,103],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,14],[2,14],[11,14],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[4,5,7,103]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0081","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"values\":[5,6,8,104],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094081,"parameters":{"masses":[[1,15],[2,15],[12,15],[0,1]],"values":[5,6,8,104],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,15],[2,15],[12,15],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[5,6,8,104]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0082","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"values\":[6,7,9,105],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094082,"parameters":{"masses":[[1,16],[2,16],[13,16],[0,1]],"values":[6,7,9,105],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,16],[2,16],[13,16],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[6,7,9,105]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0083","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"values\":[7,8,10,106],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094083,"parameters":{"masses":[[1,17],[2,17],[14,17],[0,1]],"values":[7,8,10,106],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,17],[2,17],[14,17],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[7,8,10,106]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0084","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"values\":[8,9,11,107],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094084,"parameters":{"masses":[[1,18],[2,18],[15,18],[0,1]],"values":[8,9,11,107],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,18],[2,18],[15,18],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[8,9,11,107]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0085","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"values\":[9,10,12,108],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094085,"parameters":{"masses":[[1,19],[2,19],[16,19],[0,1]],"values":[9,10,12,108],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,19],[2,19],[16,19],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[9,10,12,108]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0086","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"values\":[10,11,13,109],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094086,"parameters":{"masses":[[1,20],[2,20],[17,20],[0,1]],"values":[10,11,13,109],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,20],[2,20],[17,20],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[10,11,13,109]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0087","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"values\":[11,12,14,110],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094087,"parameters":{"masses":[[1,21],[2,21],[18,21],[0,1]],"values":[11,12,14,110],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,21],[2,21],[18,21],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[11,12,14,110]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0088","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"values\":[12,13,15,111],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094088,"parameters":{"masses":[[1,22],[2,22],[19,22],[0,1]],"values":[12,13,15,111],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,22],[2,22],[19,22],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[12,13,15,111]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0089","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"values\":[13,14,16,112],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094089,"parameters":{"masses":[[1,23],[2,23],[20,23],[0,1]],"values":[13,14,16,112],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,23],[2,23],[20,23],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[13,14,16,112]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0090","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"values\":[14,15,17,113],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094090,"parameters":{"masses":[[1,24],[2,24],[21,24],[0,1]],"values":[14,15,17,113],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,24],[2,24],[21,24],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[14,15,17,113]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0091","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"values\":[15,16,18,114],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094091,"parameters":{"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,25],[2,25],[22,25],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[15,16,18,114]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0092","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"values\":[16,17,19,115],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094092,"parameters":{"masses":[[1,26],[2,26],[23,26],[0,1]],"values":[16,17,19,115],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,26],[2,26],[23,26],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[16,17,19,115]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0093","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"values\":[17,18,20,116],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094093,"parameters":{"masses":[[1,27],[2,27],[24,27],[0,1]],"values":[17,18,20,116],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,27],[2,27],[24,27],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[17,18,20,116]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0094","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"values\":[18,19,21,117],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094094,"parameters":{"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,28],[2,28],[25,28],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[18,19,21,117]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0095","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"values\":[19,20,22,118],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094095,"parameters":{"masses":[[1,29],[2,29],[26,29],[0,1]],"values":[19,20,22,118],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,29],[2,29],[26,29],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[19,20,22,118]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0096","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"values\":[20,21,23,119],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094096,"parameters":{"masses":[[1,30],[2,30],[27,30],[0,1]],"values":[20,21,23,119],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,30],[2,30],[27,30],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[20,21,23,119]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0097","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"values\":[21,22,24,120],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094097,"parameters":{"masses":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,31],[2,31],[28,31],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[21,22,24,120]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0098","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"values\":[22,23,25,121],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094098,"parameters":{"masses":[[1,32],[2,32],[29,32],[0,1]],"values":[22,23,25,121],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,32],[2,32],[29,32],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[22,23,25,121]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0099","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"values\":[23,24,26,122],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094099,"parameters":{"masses":[[1,33],[2,33],[30,33],[0,1]],"values":[23,24,26,122],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,33],[2,33],[30,33],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[23,24,26,122]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261109-r04-law-of-total-expectation-0100","batch_id":"batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-law-of-total-expectation"],"title_zh":"全期望公式","stem_zh":"同一有限probability space上的完整disjoint partition與integrable X。操作=totalexp-zero-mass-boundary；資料={\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"values\":[24,25,27,123],\"partition\":[[0,1,2],[3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)","explanation_zh":"Wave6第三層核對=([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, 是, 否, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)；依完整明示資料獨立重算；positive cells用P(B)E[X|B]加權且tower expectation等於E[X]；zero-mass cell禁止division且weighted contribution為0；不要求independence也不混同Bayes。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["全期望公式"]},"variant":{"family_id":"qf-mod40-totalexp-zero-mass-boundary","seed":202611094100,"parameters":{"masses":[[1,34],[2,34],[31,34],[0,1]],"values":[24,25,27,123],"partition":[[0,1,2],[3]],"domain":"Finite_partition_total_expectation","operation":"totalexp-zero-mass-boundary","semantic_template":"qf-mod40-totalexp-zero-mass-boundary/{\"masses\":[[1,34],[2,34],[31,34],[0,1]],\"partition\":[[0,1,2],[3]],\"values\":[24,25,27,123]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、single-element stabilizer、finite-prefix BW、single-cover compactness、zero-mass division、noncommutative binomial、incomplete Möbius factorization、bounded-search nonmembership、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}