{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0001","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41e361d43f17734f261cf8889a28da455603f884d4515e00a2da6c4ee32c4364","family_id":"qf-density11-sample-ordered-product","seed":202610116001,"semantic_template":"qf-density11-sample-ordered-product/[[0],[0]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[0], second-stage outcomes B=[0]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[0, 0]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[0, 0]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0002","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e35303c4f382c17f54c7bfc5b228818106884451de948801ab3c47764d5e3960","family_id":"qf-density11-sample-ordered-product","seed":202610116002,"semantic_template":"qf-density11-sample-ordered-product/[[-1,1],[0,2]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-1, 1], second-stage outcomes B=[0, 2]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-1, 0], [-1, 2], [1, 0], [1, 2]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-1, 0], [-1, 2], [1, 0], [1, 2]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0003","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f2a0a92648f4c49b16bca101fc4583d2f87a115b9a2e5bc4dccaef834803f53","family_id":"qf-density11-sample-ordered-product","seed":202610116003,"semantic_template":"qf-density11-sample-ordered-product/[[-2,2],[-1,0,3]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-2, 2], second-stage outcomes B=[-1, 0, 3]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-2, -1], [-2, 0], [-2, 3], [2, -1], [2, 0], [2, 3]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-2, -1], [-2, 0], [-2, 3], [2, -1], [2, 0], [2, 3]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0004","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3df86be6b329c5df4731a64ee8f2797ed1538afe1e9a06f80e1f9664fc66232d","family_id":"qf-density11-sample-ordered-product","seed":202610116004,"semantic_template":"qf-density11-sample-ordered-product/[[-3,3],[0,4]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-3, 3], second-stage outcomes B=[0, 4]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-3, 0], [-3, 4], [3, 0], [3, 4]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-3, 0], [-3, 4], [3, 0], [3, 4]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0005","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97ec7a806e2f348293ed12e113ae3c017bb41119e4471f7bc289f7f9a564c7a4","family_id":"qf-density11-sample-ordered-product","seed":202610116005,"semantic_template":"qf-density11-sample-ordered-product/[[-4,4],[-1,0,5]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-4, 4], second-stage outcomes B=[-1, 0, 5]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-4, -1], [-4, 0], [-4, 5], [4, -1], [4, 0], [4, 5]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-4, -1], [-4, 0], [-4, 5], [4, -1], [4, 0], [4, 5]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0006","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f637c66e2a0d9c1b4ee843e3ac6e05b95efceaf091d564a83a1ca8efd4c8691","family_id":"qf-density11-sample-ordered-product","seed":202610116006,"semantic_template":"qf-density11-sample-ordered-product/[[-5,5],[0,6]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-5, 5], second-stage outcomes B=[0, 6]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-5, 0], [-5, 6], [5, 0], [5, 6]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-5, 0], [-5, 6], [5, 0], [5, 6]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0007","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7494761bbecac387ac53dff49678a0a9a0dc9e56941ad72903b40a9e7b34c358","family_id":"qf-density11-sample-ordered-product","seed":202610116007,"semantic_template":"qf-density11-sample-ordered-product/[[-6,6],[-1,0,7]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-6, 6], second-stage outcomes B=[-1, 0, 7]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-6, -1], [-6, 0], [-6, 7], [6, -1], [6, 0], [6, 7]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-6, -1], [-6, 0], [-6, 7], [6, -1], [6, 0], [6, 7]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0008","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c337ab6387f30b3325344e583df47bac21778bbec2f2709cfb1aab8008c8032f","family_id":"qf-density11-sample-ordered-product","seed":202610116008,"semantic_template":"qf-density11-sample-ordered-product/[[-7,7],[0,8]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-7, 7], second-stage outcomes B=[0, 8]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-7, 0], [-7, 8], [7, 0], [7, 8]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-7, 0], [-7, 8], [7, 0], [7, 8]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0009","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4ba0c8855afeeec4fa5c79c627d9395c2fbdef7c2dab7267fcb8fa9fb5aa3b2","family_id":"qf-density11-sample-ordered-product","seed":202610116009,"semantic_template":"qf-density11-sample-ordered-product/[[-8,8],[-1,0,9]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-8, 8], second-stage outcomes B=[-1, 0, 9]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-8, -1], [-8, 0], [-8, 9], [8, -1], [8, 0], [8, 9]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-8, -1], [-8, 0], [-8, 9], [8, -1], [8, 0], [8, 9]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0010","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06a3c54ea9b084b8bdb739d22ea6ec3801e8a463f135d63e1c0111046ab57114","family_id":"qf-density11-sample-ordered-product","seed":202610116010,"semantic_template":"qf-density11-sample-ordered-product/[[-9,9],[0,10]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-9, 9], second-stage outcomes B=[0, 10]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-9, 0], [-9, 10], [9, 0], [9, 10]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-9, 0], [-9, 10], [9, 0], [9, 10]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0011","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6adc6edf239fcb1f840180e883e91581556124f056263dfa4737726f7949d223","family_id":"qf-density11-sample-ordered-product","seed":202610116011,"semantic_template":"qf-density11-sample-ordered-product/[[-10,10],[-1,0,11]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-10, 10], second-stage outcomes B=[-1, 0, 11]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-10, -1], [-10, 0], [-10, 11], [10, -1], [10, 0], [10, 11]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-10, -1], [-10, 0], [-10, 11], [10, -1], [10, 0], [10, 11]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0012","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0aa25b8087617c62926b1e6cc64c09f7fc6f809c8d77a3d2e36945694f8b9134","family_id":"qf-density11-sample-ordered-product","seed":202610116012,"semantic_template":"qf-density11-sample-ordered-product/[[-11,11],[0,12]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-11, 11], second-stage outcomes B=[0, 12]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-11, 0], [-11, 12], [11, 0], [11, 12]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-11, 0], [-11, 12], [11, 0], [11, 12]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0013","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34e173942ce5577ae4b12b4b07d6e81663feec4fbdd31efa728aac543989ee35","family_id":"qf-density11-sample-ordered-product","seed":202610116013,"semantic_template":"qf-density11-sample-ordered-product/[[-12,12],[-1,0,13]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-12, 12], second-stage outcomes B=[-1, 0, 13]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-12, -1], [-12, 0], [-12, 13], [12, -1], [12, 0], [12, 13]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-12, -1], [-12, 0], [-12, 13], [12, -1], [12, 0], [12, 13]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0014","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3eaa4ebcfd6ca0ca8abe264a6c0ca4fa9e2933360b671a7606b099e00600669b","family_id":"qf-density11-sample-ordered-product","seed":202610116014,"semantic_template":"qf-density11-sample-ordered-product/[[-13,13],[0,14]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-13, 13], second-stage outcomes B=[0, 14]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-13, 0], [-13, 14], [13, 0], [13, 14]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-13, 0], [-13, 14], [13, 0], [13, 14]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0015","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60bac912ce67dc7d313f0d7fc93694dea94e6df1996cb79ccad2ef54b4510fac","family_id":"qf-density11-sample-ordered-product","seed":202610116015,"semantic_template":"qf-density11-sample-ordered-product/[[-14,14],[-1,0,15]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-14, 14], second-stage outcomes B=[-1, 0, 15]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-14, -1], [-14, 0], [-14, 15], [14, -1], [14, 0], [14, 15]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-14, -1], [-14, 0], [-14, 15], [14, -1], [14, 0], [14, 15]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0016","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0f6d854b41c3ad41baed132284d63249d5fa7421315c14cdd74ed4be904465cc","family_id":"qf-density11-sample-ordered-product","seed":202610116016,"semantic_template":"qf-density11-sample-ordered-product/[[-15,15],[0,16]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-15, 15], second-stage outcomes B=[0, 16]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-15, 0], [-15, 16], [15, 0], [15, 16]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-15, 0], [-15, 16], [15, 0], [15, 16]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0017","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c328653741ebad4d54a45965e0fde4ea191e238e99e6d4692110604411ddf570","family_id":"qf-density11-sample-ordered-product","seed":202610116017,"semantic_template":"qf-density11-sample-ordered-product/[[-16,16],[-1,0,17]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-16, 16], second-stage outcomes B=[-1, 0, 17]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-16, -1], [-16, 0], [-16, 17], [16, -1], [16, 0], [16, 17]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-16, -1], [-16, 0], [-16, 17], [16, -1], [16, 0], [16, 17]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0018","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4b858d1434cc15ae02cb57a7f937b8c0ef131d9887d32559963b84d21b689bb","family_id":"qf-density11-sample-ordered-product","seed":202610116018,"semantic_template":"qf-density11-sample-ordered-product/[[-17,17],[0,18]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-17, 17], second-stage outcomes B=[0, 18]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-17, 0], [-17, 18], [17, 0], [17, 18]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-17, 0], [-17, 18], [17, 0], [17, 18]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0019","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d7e6f88d5a9b977dd536c2e79e351d8a2a8b9ac5d425f62d677d4edded8385e","family_id":"qf-density11-sample-ordered-product","seed":202610116019,"semantic_template":"qf-density11-sample-ordered-product/[[-18,18],[-1,0,19]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-18, 18], second-stage outcomes B=[-1, 0, 19]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-18, -1], [-18, 0], [-18, 19], [18, -1], [18, 0], [18, 19]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-18, -1], [-18, 0], [-18, 19], [18, -1], [18, 0], [18, 19]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0020","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbfce106f836af5b4767ace2a9d6598a90f73c8735a8e4d1a04e37133e559c10","family_id":"qf-density11-sample-ordered-product","seed":202610116020,"semantic_template":"qf-density11-sample-ordered-product/[[-19,19],[0,20]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-19, 19], second-stage outcomes B=[0, 20]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-19, 0], [-19, 20], [19, 0], [19, 20]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-19, 0], [-19, 20], [19, 0], [19, 20]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0021","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"374690b3d3647a1ae6ebc92779586d92f15e0f921db318e49493dcfd33ec86c7","family_id":"qf-density11-sample-ordered-product","seed":202610116021,"semantic_template":"qf-density11-sample-ordered-product/[[-20,20],[-1,0,21]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-20, 20], second-stage outcomes B=[-1, 0, 21]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-20, -1], [-20, 0], [-20, 21], [20, -1], [20, 0], [20, 21]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-20, -1], [-20, 0], [-20, 21], [20, -1], [20, 0], [20, 21]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0022","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"332a8dda2a2cd0a978f22e1be7f473d3a410d17dce810f621214761fc666588f","family_id":"qf-density11-sample-ordered-product","seed":202610116022,"semantic_template":"qf-density11-sample-ordered-product/[[-21,21],[0,22]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-21, 21], second-stage outcomes B=[0, 22]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-21, 0], [-21, 22], [21, 0], [21, 22]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-21, 0], [-21, 22], [21, 0], [21, 22]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0023","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b592d46da256ed83286628f10075cd8757bf464266bca36b610798ce150ea6fe","family_id":"qf-density11-sample-ordered-product","seed":202610116023,"semantic_template":"qf-density11-sample-ordered-product/[[-22,22],[-1,0,23]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-22, 22], second-stage outcomes B=[-1, 0, 23]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-22, -1], [-22, 0], [-22, 23], [22, -1], [22, 0], [22, 23]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-22, -1], [-22, 0], [-22, 23], [22, -1], [22, 0], [22, 23]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0024","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35c05067a273cf0006195fad14b6f582e9e33163aa65bcfdc8f68788c3fbd7a5","family_id":"qf-density11-sample-ordered-product","seed":202610116024,"semantic_template":"qf-density11-sample-ordered-product/[[-23,23],[0,24]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-23, 23], second-stage outcomes B=[0, 24]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-23, 0], [-23, 24], [23, 0], [23, 24]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-23, 0], [-23, 24], [23, 0], [23, 24]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0025","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f118a414e31178b749f7cc5048bb63ceda1a94449abc595f0a672dc4d281d93f","family_id":"qf-density11-sample-ordered-product","seed":202610116025,"semantic_template":"qf-density11-sample-ordered-product/[[-24,24],[-1,0,25]]"},"fields":{"title":"All Ordered Elementary Outcomes","stem":"First-stage outcomes A=[-24, 24], second-stage outcomes B=[-1, 0, 25]; every pair is allowed. Normalize and sort each axis, then list all ordered pairs with A as the outer loop and B as the inner loop.","answer":{"kind":"expression","value":"[[-24, -1], [-24, 0], [-24, 25], [24, -1], [24, 0], [24, 25]]","explanation":"Pair every first-stage value with every second-stage value. Order matters; no probabilities, equiprobability, or stochastic independence are asserted. The exact result, retaining the requested order, is [[-24, -1], [-24, 0], [-24, 25], [24, -1], [24, 0], [24, 25]]."},"skills":["All Ordered Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0026","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91b9f50d18907ca1b59db11ee8a4c074fd23fc94d58b8abd914d19213fe79663","family_id":"qf-density11-sample-recording-image","seed":202610116026,"semantic_template":"qf-density11-sample-recording-image/[0]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(1)={0,…,0}, with recording=[0] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([0], [[0]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([0], [[0]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0027","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f015ec1556454185e4bf3e77da894d8b142d78138b5920b99a658fcc4c279f86","family_id":"qf-density11-sample-recording-image","seed":202610116027,"semantic_template":"qf-density11-sample-recording-image/[2,1,2,1]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[2, 1, 2, 1] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([1, 2], [[1, 3], [0, 2]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([1, 2], [[1, 3], [0, 2]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0028","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4d01f5bbaf550644e558bbec52d1220a946ba68db18edb366d6d8128aa51fa4","family_id":"qf-density11-sample-recording-image","seed":202610116028,"semantic_template":"qf-density11-sample-recording-image/[-2,0,-2,2]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-2, 0, -2, 2] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-2, 0, 2], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-2, 0, 2], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0029","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5eacd93a6a9e29213e4d5f36b8d78b97548a1c3ec0ae73b0cc0c802b5114220","family_id":"qf-density11-sample-recording-image","seed":202610116029,"semantic_template":"qf-density11-sample-recording-image/[-3,0,-3,3]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-3, 0, -3, 3] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-3, 0, 3], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-3, 0, 3], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0030","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38dc6ce5d664f3886306b06c867c27f5f6e1b4df5218b4b547d36499a5a9daba","family_id":"qf-density11-sample-recording-image","seed":202610116030,"semantic_template":"qf-density11-sample-recording-image/[-4,0,-4,4]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-4, 0, -4, 4] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-4, 0, 4], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-4, 0, 4], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0031","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92d5e109dccd29a6468dfe973a2d6d2bd1560c90b16dd6bf5aff199b0a3d6e5a","family_id":"qf-density11-sample-recording-image","seed":202610116031,"semantic_template":"qf-density11-sample-recording-image/[-5,0,-5,5]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-5, 0, -5, 5] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-5, 0, 5], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-5, 0, 5], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0032","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"099acc9789296a21adcf04329c04ca9d13e1f80237455b608b54dc344ff174ae","family_id":"qf-density11-sample-recording-image","seed":202610116032,"semantic_template":"qf-density11-sample-recording-image/[-6,0,-6,6]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-6, 0, -6, 6] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-6, 0, 6], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-6, 0, 6], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0033","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5103152aec1e2ee0850529ae078e3d6f7b361901d2915717b38f7e518e0fc4cf","family_id":"qf-density11-sample-recording-image","seed":202610116033,"semantic_template":"qf-density11-sample-recording-image/[-7,0,-7,7]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-7, 0, -7, 7] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-7, 0, 7], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-7, 0, 7], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0034","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3e6cfb745f43fe53ec65e1e2520945c9b907cc07384e2ab9267dcff21a842b0","family_id":"qf-density11-sample-recording-image","seed":202610116034,"semantic_template":"qf-density11-sample-recording-image/[-8,0,-8,8]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-8, 0, -8, 8] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-8, 0, 8], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-8, 0, 8], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0035","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b629361247d683d558ffc3b17594d4693e89f9628def831f12d51f728b83adf1","family_id":"qf-density11-sample-recording-image","seed":202610116035,"semantic_template":"qf-density11-sample-recording-image/[-9,0,-9,9]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-9, 0, -9, 9] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-9, 0, 9], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-9, 0, 9], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0036","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66d658c88340bd56c31b4ce9e148791682253c492fb3aba818945c4f84acb670","family_id":"qf-density11-sample-recording-image","seed":202610116036,"semantic_template":"qf-density11-sample-recording-image/[-10,0,-10,10]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-10, 0, -10, 10] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-10, 0, 10], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-10, 0, 10], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0037","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b50a3fdddadf7252e1a3812020ee8d5f537c4a90cda9f5da49f0b59a7a65a6c","family_id":"qf-density11-sample-recording-image","seed":202610116037,"semantic_template":"qf-density11-sample-recording-image/[-11,0,-11,11]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-11, 0, -11, 11] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-11, 0, 11], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-11, 0, 11], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0038","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3ebcdbb5a23e0e191d7086b3cceaccad8f6364ff9b74ca2b60882ff8350345f","family_id":"qf-density11-sample-recording-image","seed":202610116038,"semantic_template":"qf-density11-sample-recording-image/[-12,0,-12,12]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-12, 0, -12, 12] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-12, 0, 12], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-12, 0, 12], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0039","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92fb8969f6525b2cb6060c5ff8a9c5f7605c918f148b6d2e152b8a7efed0d9cb","family_id":"qf-density11-sample-recording-image","seed":202610116039,"semantic_template":"qf-density11-sample-recording-image/[-13,0,-13,13]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-13, 0, -13, 13] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-13, 0, 13], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-13, 0, 13], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0040","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b0c073b15659751f74fb78e8737c598e4abf36a80fd488416c6fb55b0cca1da","family_id":"qf-density11-sample-recording-image","seed":202610116040,"semantic_template":"qf-density11-sample-recording-image/[-14,0,-14,14]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-14, 0, -14, 14] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-14, 0, 14], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-14, 0, 14], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0041","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"001e5b711ac57f1fc12f33dd517afde0fd1e2f0caa45789a23d3c97045455ed8","family_id":"qf-density11-sample-recording-image","seed":202610116041,"semantic_template":"qf-density11-sample-recording-image/[-15,0,-15,15]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-15, 0, -15, 15] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-15, 0, 15], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-15, 0, 15], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0042","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faa1dbd47f45efc22bd82cc1e758dacd57d13b8c1addd62d2748321b5cf3c487","family_id":"qf-density11-sample-recording-image","seed":202610116042,"semantic_template":"qf-density11-sample-recording-image/[-16,0,-16,16]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-16, 0, -16, 16] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-16, 0, 16], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-16, 0, 16], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0043","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f085bd07f45ed7f8ba891e4f402e7f6cabaf1ea997b1a6cf69857dbaade85fc3","family_id":"qf-density11-sample-recording-image","seed":202610116043,"semantic_template":"qf-density11-sample-recording-image/[-17,0,-17,17]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-17, 0, -17, 17] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-17, 0, 17], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-17, 0, 17], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0044","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"404e668139546ff25534e7b2f8bb685734dc1eb0f03c27d5d9a44481dcd98f46","family_id":"qf-density11-sample-recording-image","seed":202610116044,"semantic_template":"qf-density11-sample-recording-image/[-18,0,-18,18]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-18, 0, -18, 18] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-18, 0, 18], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-18, 0, 18], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0045","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0eafdb30d3f388dcb500db8af09d9392dc123f2d908103aabd32a49a8ed0db7b","family_id":"qf-density11-sample-recording-image","seed":202610116045,"semantic_template":"qf-density11-sample-recording-image/[-19,0,-19,19]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-19, 0, -19, 19] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-19, 0, 19], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-19, 0, 19], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0046","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"458129ad4bec79a611245bd5df120603dab63bb73f4e085e150f39737b581d9c","family_id":"qf-density11-sample-recording-image","seed":202610116046,"semantic_template":"qf-density11-sample-recording-image/[-20,0,-20,20]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-20, 0, -20, 20] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-20, 0, 20], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-20, 0, 20], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0047","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5693527f69d1ab33e10d8c05364a0994fe2de138795f01b9de2a42f3cef3790a","family_id":"qf-density11-sample-recording-image","seed":202610116047,"semantic_template":"qf-density11-sample-recording-image/[-21,0,-21,21]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-21, 0, -21, 21] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-21, 0, 21], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-21, 0, 21], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0048","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19ff6910aa3a0fbc7421a2eab6a47f8c699f31ca79d7a3e32b366c62e3c862f7","family_id":"qf-density11-sample-recording-image","seed":202610116048,"semantic_template":"qf-density11-sample-recording-image/[-22,0,-22,22]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-22, 0, -22, 22] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-22, 0, 22], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-22, 0, 22], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0049","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"787782fd43427139d59dbe78a31f1cf0849a39bea1dd4ad8ffbf4ca9c981b9ba","family_id":"qf-density11-sample-recording-image","seed":202610116049,"semantic_template":"qf-density11-sample-recording-image/[-23,0,-23,23]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-23, 0, -23, 23] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-23, 0, 23], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-23, 0, 23], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0050","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4439039c5a12b583b66c4a508eaa5c92acf9e04909ca347f79131a376dfbcc3","family_id":"qf-density11-sample-recording-image","seed":202610116050,"semantic_template":"qf-density11-sample-recording-image/[-24,0,-24,24]"},"fields":{"title":"Recording Codes and Outcome Fibers","stem":"Elementary outcomes are Fin(4)={0,…,3}, with recording=[-24, 0, -24, 24] in original index order. Give (increasing distinct recording codes, aligned original-index fiber lists), sorting each fiber increasingly.","answer":{"kind":"expression","value":"([-24, 0, 24], [[0, 2], [1], [3]])","explanation":"Group original indices by their recorded code; many-to-one recording is allowed and does not change elementary-outcome eligibility or assign probabilities. The exact result, retaining the requested order, is ([-24, 0, 24], [[0, 2], [1], [3]])."},"skills":["Recording Codes and Outcome Fibers"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0051","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c4da221e48cd309ac6d472e4ac5eaa89721e60944f33b27b69b16d3a7b09495","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116051,"semantic_template":"qf-density11-sample-list-completeness-defects/[[],[]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(1)={0,…,0}. Candidate raw list=[0]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([], [])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([], [])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0052","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f272ecbd9bd92ef67310a30779280036cd30925565cc0d14d78158e681e92a5e","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116052,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(1)={0,…,0}. Candidate raw list=[]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0053","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d90c6bac5e26a92d3bf80371835e4a325b4d462bc59d744e17b6c300f1067a","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116053,"semantic_template":"qf-density11-sample-list-completeness-defects/[[],[100]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(3)={0,…,2}. Candidate raw list=[0, 1, 2, 100]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([], [100])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([], [100])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0054","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff8fd1ded2c3b8a91497ecd3fa861c3be3a821d7d8ee5efe05963e333aa4ae8c","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116054,"semantic_template":"qf-density11-sample-list-completeness-defects/[[3],[-4]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(4)={0,…,3}. Candidate raw list=[0, 1, 2, -4]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([3], [-4])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([3], [-4])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0055","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0dbecbab7a63ccf9e8f698ab658cf08ac08f038b76908da6d788e9b9dfe5b27","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116055,"semantic_template":"qf-density11-sample-list-completeness-defects/[[4],[-5]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(5)={0,…,4}. Candidate raw list=[0, 1, 2, 3, -5]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([4], [-5])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([4], [-5])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0056","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"397f94244fa19008a25327e8da3a971fb8534b5cbf0222ce58aca88428c38219","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116056,"semantic_template":"qf-density11-sample-list-completeness-defects/[[5],[-6]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(6)={0,…,5}. Candidate raw list=[0, 1, 2, 3, 4, -6]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([5], [-6])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([5], [-6])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0057","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b315ad8ed934f0676646d001d24562ddc07bc8f1c93489a1491cff0b6ac9d413","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116057,"semantic_template":"qf-density11-sample-list-completeness-defects/[[6],[-7]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(7)={0,…,6}. Candidate raw list=[0, 1, 2, 3, 4, 5, -7]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([6], [-7])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([6], [-7])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0058","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80a04b65d62aec64c65e861b02004932dcfed77fe85096a918f40cc35d847971","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116058,"semantic_template":"qf-density11-sample-list-completeness-defects/[[7],[-8]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(8)={0,…,7}. Candidate raw list=[0, 1, 2, 3, 4, 5, 6, -8]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([7], [-8])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([7], [-8])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0059","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87dc3bc13dd7fcd28d0d0f249e96d9897d284506ba228494f2ccc5fb93f925b0","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116059,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-9]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(1)={0,…,0}. Candidate raw list=[-9]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-9])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-9])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0060","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b61c268c1d5ba652150b3eeb0e938d9d5485e37f0ee4bcfefb9e892ec89eb5","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116060,"semantic_template":"qf-density11-sample-list-completeness-defects/[[1],[-10]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(2)={0,…,1}. Candidate raw list=[0, -10]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [-10])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([1], [-10])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0061","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"593ee41dc18decabda682f719ac8ad5794c31dab012a23399d5558d2e99f8c23","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116061,"semantic_template":"qf-density11-sample-list-completeness-defects/[[1],[-11]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(3)={0,…,2}. Candidate raw list=[0, 2, -11]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [-11])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([1], [-11])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0062","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c446d66c4fe2158e2bf12d92c7ad98cdf0dac6f7f87a69467cc0537547266d3","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116062,"semantic_template":"qf-density11-sample-list-completeness-defects/[[3],[-12]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(4)={0,…,3}. Candidate raw list=[0, 1, 2, -12]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([3], [-12])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([3], [-12])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0063","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15c45fd4dcfa1d546380528478a67ab32cacbb30a972b608b7e2157b704d7897","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116063,"semantic_template":"qf-density11-sample-list-completeness-defects/[[2],[-13]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(5)={0,…,4}. Candidate raw list=[0, 1, 3, 4, -13]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([2], [-13])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([2], [-13])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0064","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d1336384ee41e78577e2a34eee344f544c526a86bfe77b38b62220dbc3dabcf","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116064,"semantic_template":"qf-density11-sample-list-completeness-defects/[[1],[-14]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(6)={0,…,5}. Candidate raw list=[0, 2, 3, 4, 5, -14]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [-14])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([1], [-14])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0065","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d92a5589da9b4e1753822d0d055cb02f90303a1fb9183f6d23b12ad133ed2a5c","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116065,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-15]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(7)={0,…,6}. Candidate raw list=[1, 2, 3, 4, 5, 6, -15]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-15])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-15])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0066","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69314534adb07c0145084207df7a7730285b3d90e380d81610063de184b79010","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116066,"semantic_template":"qf-density11-sample-list-completeness-defects/[[7],[-16]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(8)={0,…,7}. Candidate raw list=[0, 1, 2, 3, 4, 5, 6, -16]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([7], [-16])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([7], [-16])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0067","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5063c712f3e8fd1473cf362a955f1ce14afb27719c4ebbee005c6fe2aff59147","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116067,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-17]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(1)={0,…,0}. Candidate raw list=[-17]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-17])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-17])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0068","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f383d06b81da8c894204c24902cece5aaa3645c47cb09b8236a97a5d660b1769","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116068,"semantic_template":"qf-density11-sample-list-completeness-defects/[[1],[-18]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(2)={0,…,1}. Candidate raw list=[0, -18]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [-18])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([1], [-18])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0069","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"026df48252306b236e0eb316bdbc5f6811225dd6cb82059c1151ed683cd43318","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116069,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-19]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(3)={0,…,2}. Candidate raw list=[1, 2, -19]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-19])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-19])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0070","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10026f479a49ef2285dcb69a1cdc52577cf4baa08cc00c6e02366e062ae1b8ee","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116070,"semantic_template":"qf-density11-sample-list-completeness-defects/[[3],[-20]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(4)={0,…,3}. Candidate raw list=[0, 1, 2, -20]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([3], [-20])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([3], [-20])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0071","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a704195debb1d36528e47418f6a01ee9f6f534874ceea65f504e0aa243f749c8","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116071,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-21]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(5)={0,…,4}. Candidate raw list=[1, 2, 3, 4, -21]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-21])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-21])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0072","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"225f08ef1b9570ced7710a62ed0c23541a9978f6e9c010f793ba0d5d2fbb1853","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116072,"semantic_template":"qf-density11-sample-list-completeness-defects/[[3],[-22]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(6)={0,…,5}. Candidate raw list=[0, 1, 2, 4, 5, -22]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([3], [-22])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([3], [-22])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0073","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3ff32a85858b187802ce7e3be7e99ceca51329787c0ca797c64d72d8439c344","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116073,"semantic_template":"qf-density11-sample-list-completeness-defects/[[1],[-23]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(7)={0,…,6}. Candidate raw list=[0, 2, 3, 4, 5, 6, -23]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [-23])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([1], [-23])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0074","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a45adb46781ffea4f36bc6ca98e298fa5413d91d30d4f0fbe6a03887f82beee3","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116074,"semantic_template":"qf-density11-sample-list-completeness-defects/[[7],[-24]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(8)={0,…,7}. Candidate raw list=[0, 1, 2, 3, 4, 5, 6, -24]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([7], [-24])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([7], [-24])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0075","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"286aac86f4b7b36230223e3101964be9f2b8e906e6353f0b83a9bd7b8678c548","family_id":"qf-density11-sample-list-completeness-defects","seed":202610116075,"semantic_template":"qf-density11-sample-list-completeness-defects/[[0],[-25]]"},"fields":{"title":"Missing and Outside Elementary Outcomes","stem":"Elementary outcomes are Fin(1)={0,…,0}. Candidate raw list=[-25]. Give (missing elementary outcomes, listed values outside the space), deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [-25])","explanation":"Compute the two directional differences; repetition cannot replace a missing outcome, and outside values are not elementary outcomes. No probabilities are specified. The exact result, retaining the requested order, is ([0], [-25])."},"skills":["Missing and Outside Elementary Outcomes"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0076","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"389023cae4e229e4120033ca350fd1278f2743352f9ca59849ce1644f4498a54","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116076,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"outside_space\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(1)={0,…,0}, event E=[] is a subset of the space, and ordered candidates=[100, 101, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, No, No], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, No, No], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0077","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a8fa7064b4c759b0b435500da3047a0055c8756eee20862edabc4d591895b04","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116077,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_space\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 101, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, No, No], [Yes, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, No, No], [Yes, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0078","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24c194b2e3bc294a3106716dd7519e574b606496ec55a665da2335dd43f920f7","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116078,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"outside_event\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 4, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, No], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0079","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dfc1b6de009cb6f22bdb1aad33910cb9a8744bc0613396ca6365f56dd3b0082","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116079,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_event\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 4, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0080","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e044636e018555839dc436d414d9150b83bf06232248475ca5d9f871584648b3","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116080,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"in_event\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 2, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, Yes, No], [No, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, Yes, No], [No, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0081","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa4f8166d1b8d865eb3b50251e00517c8222ba1dba2758812631e7cd2d2fd71f","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116081,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"in_event\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 2, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, No], [No, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, No], [No, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0082","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24436c0837f7b695d9d4396b84a86e69d9643c9a89ba55ec54d4d02901c24453","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116082,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"in_event\",\"outside_space\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 2, 102]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, No], [Yes, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, No], [Yes, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0083","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd4d63e7c443a142b80eb0b0466dde8263fd5552add696280120a4e1bd50867f","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116083,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"outside_space\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 101, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, No, Yes], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0084","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"757753b98642d6423bad699076052ab6a294af1c172eab13dcd20de60cdd6590","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116084,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"outside_space\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 101, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, No, Yes], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0085","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c7341d22641fe39f15184347f2da59c62a98feb7ee75d15945804514bf45c1c","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116085,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_space\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 101, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0086","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e2d2d73bc1f0191a47f1c323065883e82f670c6f9d59324c17b1e105cb0b280","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116086,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"outside_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 4, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0087","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"badbd917d02897e03a12552f647ee95dfbd9eae59b67037347007da46f065e6f","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116087,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"outside_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 4, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0088","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a317a347c26eef80b11d4768980a6df4165dc0d0299a652128d4ca93f27132a","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116088,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 4, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0089","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34a8d9c991e866bc5b1a1c675250d52e56ed933aa56f31594a6fce207bf0d5e8","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116089,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"in_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 2, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0090","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d26dfba4e3564c835535fc976967d1201ca77f90df5e3419525bc4f4c8d828e1","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116090,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"in_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 2, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0091","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4e232a54f439e8e0b341474299a32aa4ab38f4ecceb4aa757865ba4dfc3eaa","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116091,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"in_event\",\"outside_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 2, 5]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, No])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, No])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0092","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f51cededacfdf7318816d473b69e150d9e23669fd75b60262cf5cb80bd3a0678","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116092,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"outside_space\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 101, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, No, Yes], [No, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, No, Yes], [No, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0093","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fbe51578baa391a9e435d8be71e002bc8fbab8a2c7096f8f839264c676ed32c","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116093,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"outside_space\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 101, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, No, Yes], [No, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, No, Yes], [No, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0094","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e530e0a0f0201cd4d11ee46b4834cd3755909a66ec34f3e683ca0bc121f105d","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116094,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_space\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 101, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, No, Yes], [Yes, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, No, Yes], [Yes, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0095","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02804aeda35f43785b217a2f9c131c42d4f36e8d7f9a57f548644cf366b5a9e4","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116095,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"outside_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 4, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0096","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb69826b3c8bbcc136d260b100aba30331212f32306f73a751e8fbd2cdd86f14","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116096,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"outside_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 4, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0097","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1fa568e2162666803cb315b4201365c5bfd61619c8b1633d48a22b499017923","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116097,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"outside_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[1, 4, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, No, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, No, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0098","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"892660035ffaa1a1011139f1631e28371aa28d1f6d326574a452d0a3c24b63e6","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116098,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_space\",\"in_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[100, 2, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([No, Yes, Yes], [No, Yes, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([No, Yes, Yes], [No, Yes, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0099","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7035fbe22e7705b6141165548634aec9f7bccfb9a518dc1553cbd4bbacfbe84","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116099,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"outside_event\",\"in_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(6)={0,…,5}, event E=[1, 2, 3] is a subset of the space, and ordered candidates=[0, 2, 3]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [No, Yes, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [No, Yes, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261011-r06-sample-space-0100","batch_id":"batch-20261008-advance-20261011-r06-sample-space-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a02119ca0c314e633ef3475a69afcde75d0096027f844e6e3a9fd7596ebfceb","family_id":"qf-density11-sample-outcome-event-masks","seed":202610116100,"semantic_template":"qf-density11-sample-outcome-event-masks/[\"in_event\",\"in_event\",\"in_event\"]"},"fields":{"title":"Outcome and Event Membership Masks","stem":"Elementary outcomes are Fin(3)={0,…,2}, event E=[0, 1, 2] is a subset of the space, and ordered candidates=[0, 1, 2]. Give (membership-in-space mask, membership-in-E mask) using Yes/No.","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [Yes, Yes, Yes])","explanation":"Check the fixed sample space and the event separately; an event is not a single outcome, outside candidates are only inputs to be checked, and no probabilities are assigned. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [Yes, Yes, Yes])."},"skills":["Outcome and Event Membership Masks"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}