{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0001","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-12, -11}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E=∅ 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037001,"parameters":{"S":[-12,-11],"E":[],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-12,-11],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0002","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-11, -10, -9}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-11, -10, -9} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037002,"parameters":{"S":[-11,-10,-9],"E":[-11,-10,-9],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-11,-10,-9],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0003","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-10, -9, -8, -7}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-10} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/4","explanation_zh":"P(E)=1/4。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037003,"parameters":{"S":[-10,-9,-8,-7],"E":[-10],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-10],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8,-7],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0004","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-9, -8, -7, -6, -5}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-5} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/5","explanation_zh":"P(E)=1/5。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037004,"parameters":{"S":[-9,-8,-7,-6,-5],"E":[-5],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-5],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7,-6,-5],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0005","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-8, -7, -6, -5, -4, -3}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-8, -6, -4} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037005,"parameters":{"S":[-8,-7,-6,-5,-4,-3],"E":[-8,-6,-4],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-8,-6,-4],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6,-5,-4,-3],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0006","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-7, -6}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-7, -6} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037006,"parameters":{"S":[-7,-6],"E":[-7,-6],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-7,-6],\"F\":\"powerset(S)\",\"S\":[-7,-6],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0007","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-6, -5, -4}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E=∅ 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037007,"parameters":{"S":[-6,-5,-4],"E":[],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0008","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-5, -4, -3, -2}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-5, -4, -3, -2} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037008,"parameters":{"S":[-5,-4,-3,-2],"E":[-5,-4,-3,-2],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-5,-4,-3,-2],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3,-2],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0009","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-4, -3, -2, -1, 0}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-4} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/5","explanation_zh":"P(E)=1/5。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037009,"parameters":{"S":[-4,-3,-2,-1,0],"E":[-4],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-4],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2,-1,0],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0010","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-3, -2, -1, 0, 1, 2}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={2} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/6","explanation_zh":"P(E)=1/6。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037010,"parameters":{"S":[-3,-2,-1,0,1,2],"E":[2],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[2],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1,0,1,2],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0011","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-2, -1}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-2} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037011,"parameters":{"S":[-2,-1],"E":[-2],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-2],\"F\":\"powerset(S)\",\"S\":[-2,-1],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0012","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={-1, 0, 1}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={-1, 0} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"2/3","explanation_zh":"P(E)=2/3。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037012,"parameters":{"S":[-1,0,1],"E":[-1,0],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[-1,0],\"F\":\"powerset(S)\",\"S\":[-1,0,1],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0013","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={0, 1, 2, 3}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E=∅ 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037013,"parameters":{"S":[0,1,2,3],"E":[],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0014","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={1, 2, 3, 4, 5}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={1, 2, 3, 4, 5} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037014,"parameters":{"S":[1,2,3,4,5],"E":[1,2,3,4,5],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[1,2,3,4,5],\"F\":\"powerset(S)\",\"S\":[1,2,3,4,5],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0015","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={2, 3, 4, 5, 6, 7}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={2} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/6","explanation_zh":"P(E)=1/6。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037015,"parameters":{"S":[2,3,4,5,6,7],"E":[2],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[2],\"F\":\"powerset(S)\",\"S\":[2,3,4,5,6,7],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0016","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={3, 4}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={4} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037016,"parameters":{"S":[3,4],"E":[4],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[4],\"F\":\"powerset(S)\",\"S\":[3,4],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0017","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={4, 5, 6}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={4, 6} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"2/3","explanation_zh":"P(E)=2/3。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037017,"parameters":{"S":[4,5,6],"E":[4,6],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[4,6],\"F\":\"powerset(S)\",\"S\":[4,5,6],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0018","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={5, 6, 7, 8}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={5, 6} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037018,"parameters":{"S":[5,6,7,8],"E":[5,6],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[5,6],\"F\":\"powerset(S)\",\"S\":[5,6,7,8],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0019","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={6, 7, 8, 9, 10}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E=∅ 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037019,"parameters":{"S":[6,7,8,9,10],"E":[],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[6,7,8,9,10],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0020","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={7, 8, 9, 10, 11, 12}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={7, 8, 9, 10, 11, 12} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037020,"parameters":{"S":[7,8,9,10,11,12],"E":[7,8,9,10,11,12],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[7,8,9,10,11,12],\"F\":\"powerset(S)\",\"S\":[7,8,9,10,11,12],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0021","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={8, 9}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={8} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037021,"parameters":{"S":[8,9],"E":[8],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[8],\"F\":\"powerset(S)\",\"S\":[8,9],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0022","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={9, 10, 11}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={11} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/3","explanation_zh":"P(E)=1/3。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037022,"parameters":{"S":[9,10,11],"E":[11],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[11],\"F\":\"powerset(S)\",\"S\":[9,10,11],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0023","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={10, 11, 12, 13}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={10, 12} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(E)=1/2。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037023,"parameters":{"S":[10,11,12,13],"E":[10,12],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[10,12],\"F\":\"powerset(S)\",\"S\":[10,11,12,13],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0024","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={11, 12, 13, 14, 15}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E={11, 12} 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"2/5","explanation_zh":"P(E)=2/5。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037024,"parameters":{"S":[11,12,13,14,15],"E":[11,12],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[11,12],\"F\":\"powerset(S)\",\"S\":[11,12,13,14,15],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0025","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"明示等可能的事件機率","stem_zh":"非空有限樣本空間 S={12, 13, 14, 15, 16, 17}，事件族 F=𝒫(S)，所有基本結果明示等可能。事件 E=∅ 的精確 P(E) 是多少？","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。因為已明示等可能，才可用 |E|/|S|；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["明示等可能的事件機率"]},"variant":{"family_id":"qf-probability-uniform-event","seed":202610037025,"parameters":{"S":[12,13,14,15,16,17],"E":[],"F":"powerset(S)","equiprobable":true,"operation":"uniform_event","semantic_template":"qf-probability-uniform-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[12,13,14,15,16,17],\"equiprobable\":true,\"operation\":\"uniform_event\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0026","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-12, -11, -10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/2, 1/2, 0/2]，事件 E={-10}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037026,"parameters":{"S":[-12,-11,-10],"F":"powerset(S)","weights":[1,1,0],"denominator":2,"operation":"weighted_event","E":[-10],"semantic_template":"qf-probability-weighted-event/{\"E\":[-10],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10],\"denominator\":\"2\",\"operation\":\"weighted_event\",\"weights\":[[-12,\"1\"],[-11,\"1\"],[-10,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0027","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-11, -10, -9}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/5, 2/5, 1/5]，事件 E={-11, -10, -9}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037027,"parameters":{"S":[-11,-10,-9],"F":"powerset(S)","weights":[2,2,1],"denominator":5,"operation":"weighted_event","E":[-11,-10,-9],"semantic_template":"qf-probability-weighted-event/{\"E\":[-11,-10,-9],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9],\"denominator\":\"5\",\"operation\":\"weighted_event\",\"weights\":[[-11,\"2\"],[-10,\"2\"],[-9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0028","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-10, -9, -8}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/8, 3/8, 2/8]，事件 E={-10}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"3/8","explanation_zh":"P(E)=3/8。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037028,"parameters":{"S":[-10,-9,-8],"F":"powerset(S)","weights":[3,3,2],"denominator":8,"operation":"weighted_event","E":[-10],"semantic_template":"qf-probability-weighted-event/{\"E\":[-10],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8],\"denominator\":\"8\",\"operation\":\"weighted_event\",\"weights\":[[-10,\"3\"],[-9,\"3\"],[-8,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0029","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-9, -8, -7}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [4/8, 4/8, 0/8]，事件 E={-7}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037029,"parameters":{"S":[-9,-8,-7],"F":"powerset(S)","weights":[4,4,0],"denominator":8,"operation":"weighted_event","E":[-7],"semantic_template":"qf-probability-weighted-event/{\"E\":[-7],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7],\"denominator\":\"2\",\"operation\":\"weighted_event\",\"weights\":[[-9,\"1\"],[-8,\"1\"],[-7,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0030","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-8, -7, -6}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [5/11, 5/11, 1/11]，事件 E={-8, -7}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"10/11","explanation_zh":"P(E)=10/11。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037030,"parameters":{"S":[-8,-7,-6],"F":"powerset(S)","weights":[5,5,1],"denominator":11,"operation":"weighted_event","E":[-8,-7],"semantic_template":"qf-probability-weighted-event/{\"E\":[-8,-7],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6],\"denominator\":\"11\",\"operation\":\"weighted_event\",\"weights\":[[-8,\"5\"],[-7,\"5\"],[-6,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0031","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-7, -6, -5}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [6/9, 1/9, 2/9]，事件 E=∅。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037031,"parameters":{"S":[-7,-6,-5],"F":"powerset(S)","weights":[6,1,2],"denominator":9,"operation":"weighted_event","E":[],"semantic_template":"qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5],\"denominator\":\"9\",\"operation\":\"weighted_event\",\"weights\":[[-7,\"6\"],[-6,\"1\"],[-5,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0032","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-6, -5, -4}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [7/9, 2/9, 0/9]，事件 E={-6, -5, -4}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037032,"parameters":{"S":[-6,-5,-4],"F":"powerset(S)","weights":[7,2,0],"denominator":9,"operation":"weighted_event","E":[-6,-5,-4],"semantic_template":"qf-probability-weighted-event/{\"E\":[-6,-5,-4],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4],\"denominator\":\"9\",\"operation\":\"weighted_event\",\"weights\":[[-6,\"7\"],[-5,\"2\"],[-4,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0033","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-5, -4, -3}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [8/12, 3/12, 1/12]，事件 E={-5}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"2/3","explanation_zh":"P(E)=2/3。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037033,"parameters":{"S":[-5,-4,-3],"F":"powerset(S)","weights":[8,3,1],"denominator":12,"operation":"weighted_event","E":[-5],"semantic_template":"qf-probability-weighted-event/{\"E\":[-5],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3],\"denominator\":\"12\",\"operation\":\"weighted_event\",\"weights\":[[-5,\"8\"],[-4,\"3\"],[-3,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0034","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-4, -3, -2}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [9/15, 4/15, 2/15]，事件 E={-2}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"2/15","explanation_zh":"P(E)=2/15。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037034,"parameters":{"S":[-4,-3,-2],"F":"powerset(S)","weights":[9,4,2],"denominator":15,"operation":"weighted_event","E":[-2],"semantic_template":"qf-probability-weighted-event/{\"E\":[-2],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2],\"denominator\":\"15\",\"operation\":\"weighted_event\",\"weights\":[[-4,\"9\"],[-3,\"4\"],[-2,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0035","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-3, -2, -1}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [10/15, 5/15, 0/15]，事件 E={-3, -2}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037035,"parameters":{"S":[-3,-2,-1],"F":"powerset(S)","weights":[10,5,0],"denominator":15,"operation":"weighted_event","E":[-3,-2],"semantic_template":"qf-probability-weighted-event/{\"E\":[-3,-2],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1],\"denominator\":\"3\",\"operation\":\"weighted_event\",\"weights\":[[-3,\"2\"],[-2,\"1\"],[-1,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0036","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-2, -1, 0}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [11/13, 1/13, 1/13]，事件 E=∅。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037036,"parameters":{"S":[-2,-1,0],"F":"powerset(S)","weights":[11,1,1],"denominator":13,"operation":"weighted_event","E":[],"semantic_template":"qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-2,-1,0],\"denominator\":\"13\",\"operation\":\"weighted_event\",\"weights\":[[-2,\"11\"],[-1,\"1\"],[0,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0037","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={-1, 0, 1}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [12/16, 2/16, 2/16]，事件 E={-1, 0, 1}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037037,"parameters":{"S":[-1,0,1],"F":"powerset(S)","weights":[12,2,2],"denominator":16,"operation":"weighted_event","E":[-1,0,1],"semantic_template":"qf-probability-weighted-event/{\"E\":[-1,0,1],\"F\":\"powerset(S)\",\"S\":[-1,0,1],\"denominator\":\"8\",\"operation\":\"weighted_event\",\"weights\":[[-1,\"6\"],[0,\"1\"],[1,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0038","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={0, 1, 2}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [13/16, 3/16, 0/16]，事件 E={0}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"13/16","explanation_zh":"P(E)=13/16。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037038,"parameters":{"S":[0,1,2],"F":"powerset(S)","weights":[13,3,0],"denominator":16,"operation":"weighted_event","E":[0],"semantic_template":"qf-probability-weighted-event/{\"E\":[0],\"F\":\"powerset(S)\",\"S\":[0,1,2],\"denominator\":\"16\",\"operation\":\"weighted_event\",\"weights\":[[0,\"13\"],[1,\"3\"],[2,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0039","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={1, 2, 3}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [14/19, 4/19, 1/19]，事件 E={3}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/19","explanation_zh":"P(E)=1/19。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037039,"parameters":{"S":[1,2,3],"F":"powerset(S)","weights":[14,4,1],"denominator":19,"operation":"weighted_event","E":[3],"semantic_template":"qf-probability-weighted-event/{\"E\":[3],\"F\":\"powerset(S)\",\"S\":[1,2,3],\"denominator\":\"19\",\"operation\":\"weighted_event\",\"weights\":[[1,\"14\"],[2,\"4\"],[3,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0040","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={2, 3, 4}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [15/22, 5/22, 2/22]，事件 E={2, 3}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"10/11","explanation_zh":"P(E)=10/11。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037040,"parameters":{"S":[2,3,4],"F":"powerset(S)","weights":[15,5,2],"denominator":22,"operation":"weighted_event","E":[2,3],"semantic_template":"qf-probability-weighted-event/{\"E\":[2,3],\"F\":\"powerset(S)\",\"S\":[2,3,4],\"denominator\":\"22\",\"operation\":\"weighted_event\",\"weights\":[[2,\"15\"],[3,\"5\"],[4,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0041","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={3, 4, 5}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [16/17, 1/17, 0/17]，事件 E=∅。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037041,"parameters":{"S":[3,4,5],"F":"powerset(S)","weights":[16,1,0],"denominator":17,"operation":"weighted_event","E":[],"semantic_template":"qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[3,4,5],\"denominator\":\"17\",\"operation\":\"weighted_event\",\"weights\":[[3,\"16\"],[4,\"1\"],[5,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0042","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={4, 5, 6}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [17/20, 2/20, 1/20]，事件 E={4, 5, 6}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037042,"parameters":{"S":[4,5,6],"F":"powerset(S)","weights":[17,2,1],"denominator":20,"operation":"weighted_event","E":[4,5,6],"semantic_template":"qf-probability-weighted-event/{\"E\":[4,5,6],\"F\":\"powerset(S)\",\"S\":[4,5,6],\"denominator\":\"20\",\"operation\":\"weighted_event\",\"weights\":[[4,\"17\"],[5,\"2\"],[6,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0043","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={5, 6, 7}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [18/23, 3/23, 2/23]，事件 E={5}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"18/23","explanation_zh":"P(E)=18/23。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037043,"parameters":{"S":[5,6,7],"F":"powerset(S)","weights":[18,3,2],"denominator":23,"operation":"weighted_event","E":[5],"semantic_template":"qf-probability-weighted-event/{\"E\":[5],\"F\":\"powerset(S)\",\"S\":[5,6,7],\"denominator\":\"23\",\"operation\":\"weighted_event\",\"weights\":[[5,\"18\"],[6,\"3\"],[7,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0044","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={6, 7, 8}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [19/23, 4/23, 0/23]，事件 E={8}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037044,"parameters":{"S":[6,7,8],"F":"powerset(S)","weights":[19,4,0],"denominator":23,"operation":"weighted_event","E":[8],"semantic_template":"qf-probability-weighted-event/{\"E\":[8],\"F\":\"powerset(S)\",\"S\":[6,7,8],\"denominator\":\"23\",\"operation\":\"weighted_event\",\"weights\":[[6,\"19\"],[7,\"4\"],[8,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0045","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={7, 8, 9}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [20/26, 5/26, 1/26]，事件 E={7, 8}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"25/26","explanation_zh":"P(E)=25/26。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037045,"parameters":{"S":[7,8,9],"F":"powerset(S)","weights":[20,5,1],"denominator":26,"operation":"weighted_event","E":[7,8],"semantic_template":"qf-probability-weighted-event/{\"E\":[7,8],\"F\":\"powerset(S)\",\"S\":[7,8,9],\"denominator\":\"26\",\"operation\":\"weighted_event\",\"weights\":[[7,\"20\"],[8,\"5\"],[9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0046","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={8, 9, 10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [21/24, 1/24, 2/24]，事件 E=∅。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(E)=0/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037046,"parameters":{"S":[8,9,10],"F":"powerset(S)","weights":[21,1,2],"denominator":24,"operation":"weighted_event","E":[],"semantic_template":"qf-probability-weighted-event/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[8,9,10],\"denominator\":\"24\",\"operation\":\"weighted_event\",\"weights\":[[8,\"21\"],[9,\"1\"],[10,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0047","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={9, 10, 11}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [22/24, 2/24, 0/24]，事件 E={9, 10, 11}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037047,"parameters":{"S":[9,10,11],"F":"powerset(S)","weights":[22,2,0],"denominator":24,"operation":"weighted_event","E":[9,10,11],"semantic_template":"qf-probability-weighted-event/{\"E\":[9,10,11],\"F\":\"powerset(S)\",\"S\":[9,10,11],\"denominator\":\"12\",\"operation\":\"weighted_event\",\"weights\":[[9,\"11\"],[10,\"1\"],[11,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0048","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={10, 11, 12}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [23/27, 3/27, 1/27]，事件 E={10}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"23/27","explanation_zh":"P(E)=23/27。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037048,"parameters":{"S":[10,11,12],"F":"powerset(S)","weights":[23,3,1],"denominator":27,"operation":"weighted_event","E":[10],"semantic_template":"qf-probability-weighted-event/{\"E\":[10],\"F\":\"powerset(S)\",\"S\":[10,11,12],\"denominator\":\"27\",\"operation\":\"weighted_event\",\"weights\":[[10,\"23\"],[11,\"3\"],[12,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0049","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={11, 12, 13}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [24/30, 4/30, 2/30]，事件 E={13}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/15","explanation_zh":"P(E)=1/15。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037049,"parameters":{"S":[11,12,13],"F":"powerset(S)","weights":[24,4,2],"denominator":30,"operation":"weighted_event","E":[13],"semantic_template":"qf-probability-weighted-event/{\"E\":[13],\"F\":\"powerset(S)\",\"S\":[11,12,13],\"denominator\":\"15\",\"operation\":\"weighted_event\",\"weights\":[[11,\"12\"],[12,\"2\"],[13,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0050","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"非等可能加權事件機率","stem_zh":"非空有限樣本空間 S={12, 13, 14}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [25/30, 5/30, 0/30]，事件 E={12, 13}。求精確 P(E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(E)=1/1。先驗證非負權重總和為一，再相加事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["非等可能加權事件機率"]},"variant":{"family_id":"qf-probability-weighted-event","seed":202610037050,"parameters":{"S":[12,13,14],"F":"powerset(S)","weights":[25,5,0],"denominator":30,"operation":"weighted_event","E":[12,13],"semantic_template":"qf-probability-weighted-event/{\"E\":[12,13],\"F\":\"powerset(S)\",\"S\":[12,13,14],\"denominator\":\"6\",\"operation\":\"weighted_event\",\"weights\":[[12,\"5\"],[13,\"1\"],[14,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0051","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-12, -11, -10}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [1/2, 1/2, 0/2]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=2；分母=2。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037051,"parameters":{"S":[-12,-11,-10],"F":"powerset(S)","weights":[1,1,0],"denominator":2,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-12,-11,-10],\"denominator\":\"2\",\"operation\":\"model_validity\",\"weights\":[[-12,\"1\"],[-11,\"1\"],[-10,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0052","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-11, -10, -9}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-2/5, 2/5, 1/5]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=1；分母=5。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037052,"parameters":{"S":[-11,-10,-9],"F":"powerset(S)","weights":[-2,2,1],"denominator":5,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-11,-10,-9],\"denominator\":\"5\",\"operation\":\"model_validity\",\"weights\":[[-11,\"-2\"],[-10,\"2\"],[-9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0053","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-10, -9, -8}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [3/9, 3/9, 2/9]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=8；分母=9。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037053,"parameters":{"S":[-10,-9,-8],"F":"powerset(S)","weights":[3,3,2],"denominator":9,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-10,-9,-8],\"denominator\":\"9\",\"operation\":\"model_validity\",\"weights\":[[-10,\"3\"],[-9,\"3\"],[-8,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0054","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-9, -8, -7}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [4/0, 4/0, 0/0]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=8；分母=0。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037054,"parameters":{"S":[-9,-8,-7],"F":"powerset(S)","weights":[4,4,0],"denominator":0,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-9,-8,-7],\"denominator\":\"0\",\"operation\":\"model_validity\",\"weights\":[[-9,\"1\"],[-8,\"1\"],[-7,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0055","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-8, -7, -6}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [5/11, 5/11, 1/11]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=11；分母=11。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037055,"parameters":{"S":[-8,-7,-6],"F":"powerset(S)","weights":[5,5,1],"denominator":11,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-8,-7,-6],\"denominator\":\"11\",\"operation\":\"model_validity\",\"weights\":[[-8,\"5\"],[-7,\"5\"],[-6,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0056","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-7, -6, -5}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-6/9, 1/9, 2/9]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=-3；分母=9。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037056,"parameters":{"S":[-7,-6,-5],"F":"powerset(S)","weights":[-6,1,2],"denominator":9,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-7,-6,-5],\"denominator\":\"9\",\"operation\":\"model_validity\",\"weights\":[[-7,\"-6\"],[-6,\"1\"],[-5,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0057","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-6, -5, -4}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [7/10, 2/10, 0/10]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=9；分母=10。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037057,"parameters":{"S":[-6,-5,-4],"F":"powerset(S)","weights":[7,2,0],"denominator":10,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-6,-5,-4],\"denominator\":\"10\",\"operation\":\"model_validity\",\"weights\":[[-6,\"7\"],[-5,\"2\"],[-4,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0058","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-5, -4, -3}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [8/-12, 3/-12, 1/-12]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=12；分母=-12。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037058,"parameters":{"S":[-5,-4,-3],"F":"powerset(S)","weights":[8,3,1],"denominator":-12,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-5,-4,-3],\"denominator\":\"-12\",\"operation\":\"model_validity\",\"weights\":[[-5,\"8\"],[-4,\"3\"],[-3,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0059","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-4, -3, -2}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [9/15, 4/15, 2/15]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=15；分母=15。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037059,"parameters":{"S":[-4,-3,-2],"F":"powerset(S)","weights":[9,4,2],"denominator":15,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-4,-3,-2],\"denominator\":\"15\",\"operation\":\"model_validity\",\"weights\":[[-4,\"9\"],[-3,\"4\"],[-2,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0060","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-3, -2, -1}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-10/15, 5/15, 0/15]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=-5；分母=15。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037060,"parameters":{"S":[-3,-2,-1],"F":"powerset(S)","weights":[-10,5,0],"denominator":15,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-3,-2,-1],\"denominator\":\"3\",\"operation\":\"model_validity\",\"weights\":[[-3,\"-2\"],[-2,\"1\"],[-1,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0061","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-2, -1, 0}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [11/14, 1/14, 1/14]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=13；分母=14。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037061,"parameters":{"S":[-2,-1,0],"F":"powerset(S)","weights":[11,1,1],"denominator":14,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-2,-1,0],\"denominator\":\"14\",\"operation\":\"model_validity\",\"weights\":[[-2,\"11\"],[-1,\"1\"],[0,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0062","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={-1, 0, 1}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [12/-16, 2/-16, 2/-16]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=16；分母=-16。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037062,"parameters":{"S":[-1,0,1],"F":"powerset(S)","weights":[12,2,2],"denominator":-16,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[-1,0,1],\"denominator\":\"-8\",\"operation\":\"model_validity\",\"weights\":[[-1,\"6\"],[0,\"1\"],[1,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0063","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={0, 1, 2}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [13/16, 3/16, 0/16]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=16；分母=16。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037063,"parameters":{"S":[0,1,2],"F":"powerset(S)","weights":[13,3,0],"denominator":16,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[0,1,2],\"denominator\":\"16\",\"operation\":\"model_validity\",\"weights\":[[0,\"13\"],[1,\"3\"],[2,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0064","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={1, 2, 3}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-14/19, 4/19, 1/19]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=-9；分母=19。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037064,"parameters":{"S":[1,2,3],"F":"powerset(S)","weights":[-14,4,1],"denominator":19,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[1,2,3],\"denominator\":\"19\",\"operation\":\"model_validity\",\"weights\":[[1,\"-14\"],[2,\"4\"],[3,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0065","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={2, 3, 4}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [15/23, 5/23, 2/23]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=22；分母=23。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037065,"parameters":{"S":[2,3,4],"F":"powerset(S)","weights":[15,5,2],"denominator":23,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[2,3,4],\"denominator\":\"23\",\"operation\":\"model_validity\",\"weights\":[[2,\"15\"],[3,\"5\"],[4,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0066","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={3, 4, 5}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [16/-17, 1/-17, 0/-17]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=17；分母=-17。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037066,"parameters":{"S":[3,4,5],"F":"powerset(S)","weights":[16,1,0],"denominator":-17,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[3,4,5],\"denominator\":\"-17\",\"operation\":\"model_validity\",\"weights\":[[3,\"16\"],[4,\"1\"],[5,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0067","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={4, 5, 6}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [17/20, 2/20, 1/20]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=20；分母=20。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037067,"parameters":{"S":[4,5,6],"F":"powerset(S)","weights":[17,2,1],"denominator":20,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[4,5,6],\"denominator\":\"20\",\"operation\":\"model_validity\",\"weights\":[[4,\"17\"],[5,\"2\"],[6,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0068","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={5, 6, 7}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-18/23, 3/23, 2/23]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=-13；分母=23。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037068,"parameters":{"S":[5,6,7],"F":"powerset(S)","weights":[-18,3,2],"denominator":23,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[5,6,7],\"denominator\":\"23\",\"operation\":\"model_validity\",\"weights\":[[5,\"-18\"],[6,\"3\"],[7,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0069","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={6, 7, 8}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [19/24, 4/24, 0/24]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=23；分母=24。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037069,"parameters":{"S":[6,7,8],"F":"powerset(S)","weights":[19,4,0],"denominator":24,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[6,7,8],\"denominator\":\"24\",\"operation\":\"model_validity\",\"weights\":[[6,\"19\"],[7,\"4\"],[8,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0070","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={7, 8, 9}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [20/-26, 5/-26, 1/-26]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=26；分母=-26。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037070,"parameters":{"S":[7,8,9],"F":"powerset(S)","weights":[20,5,1],"denominator":-26,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[7,8,9],\"denominator\":\"-26\",\"operation\":\"model_validity\",\"weights\":[[7,\"20\"],[8,\"5\"],[9,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0071","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={8, 9, 10}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [21/24, 1/24, 2/24]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=24；分母=24。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037071,"parameters":{"S":[8,9,10],"F":"powerset(S)","weights":[21,1,2],"denominator":24,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[8,9,10],\"denominator\":\"24\",\"operation\":\"model_validity\",\"weights\":[[8,\"21\"],[9,\"1\"],[10,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0072","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={9, 10, 11}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [-22/24, 2/24, 0/24]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=-20；分母=24。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037072,"parameters":{"S":[9,10,11],"F":"powerset(S)","weights":[-22,2,0],"denominator":24,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[9,10,11],\"denominator\":\"12\",\"operation\":\"model_validity\",\"weights\":[[9,\"-11\"],[10,\"1\"],[11,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0073","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={10, 11, 12}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [23/28, 3/28, 1/28]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=27；分母=28。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037073,"parameters":{"S":[10,11,12],"F":"powerset(S)","weights":[23,3,1],"denominator":28,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[10,11,12],\"denominator\":\"28\",\"operation\":\"model_validity\",\"weights\":[[10,\"23\"],[11,\"3\"],[12,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0074","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={11, 12, 13}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [24/-30, 4/-30, 2/-30]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"否","explanation_zh":"權重分子總和=30；分母=-30。檢查正分母、非負權重與總和恰為一後答案是否；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037074,"parameters":{"S":[11,12,13],"F":"powerset(S)","weights":[24,4,2],"denominator":-30,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[11,12,13],\"denominator\":\"-15\",\"operation\":\"model_validity\",\"weights\":[[11,\"12\"],[12,\"2\"],[13,\"1\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0075","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"候選有限機率模型的合法性","stem_zh":"非空有限樣本空間 S={12, 13, 14}，事件族 F=𝒫(S)。依 S 列示順序提出候選權重 [25/30, 5/30, 0/30]，是否構成非負且總和為一的合法機率模型？","answer":{"kind":"short_text","value":"是","explanation_zh":"權重分子總和=30；分母=30。檢查正分母、非負權重與總和恰為一後答案是是；𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["候選有限機率模型的合法性"]},"variant":{"family_id":"qf-probability-model-validity","seed":202610037075,"parameters":{"S":[12,13,14],"F":"powerset(S)","weights":[25,5,0],"denominator":30,"operation":"model_validity","semantic_template":"qf-probability-model-validity/{\"F\":\"powerset(S)\",\"S\":[12,13,14],\"denominator\":\"6\",\"operation\":\"model_validity\",\"weights\":[[12,\"5\"],[13,\"1\"],[14,\"0\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0076","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-12, -11, -10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/3, 1/3, 2/3]，事件 E=∅。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037076,"parameters":{"S":[-12,-11,-10],"F":"powerset(S)","weights":[0,1,2],"denominator":3,"operation":"complement","E":[],"semantic_template":"qf-probability-complement/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-12,-11,-10],\"denominator\":\"3\",\"operation\":\"complement\",\"weights\":[[-12,\"0\"],[-11,\"1\"],[-10,\"2\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0077","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-11, -10, -9}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/6, 2/6, 3/6]，事件 E={-11, -10, -9}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(S∖E)=0/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037077,"parameters":{"S":[-11,-10,-9],"F":"powerset(S)","weights":[1,2,3],"denominator":6,"operation":"complement","E":[-11,-10,-9],"semantic_template":"qf-probability-complement/{\"E\":[-11,-10,-9],\"F\":\"powerset(S)\",\"S\":[-11,-10,-9],\"denominator\":\"6\",\"operation\":\"complement\",\"weights\":[[-11,\"1\"],[-10,\"2\"],[-9,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0078","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-10, -9, -8}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/9, 3/9, 4/9]，事件 E={-10}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"7/9","explanation_zh":"P(S∖E)=7/9。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037078,"parameters":{"S":[-10,-9,-8],"F":"powerset(S)","weights":[2,3,4],"denominator":9,"operation":"complement","E":[-10],"semantic_template":"qf-probability-complement/{\"E\":[-10],\"F\":\"powerset(S)\",\"S\":[-10,-9,-8],\"denominator\":\"9\",\"operation\":\"complement\",\"weights\":[[-10,\"2\"],[-9,\"3\"],[-8,\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0079","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-9, -8, -7}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/12, 4/12, 5/12]，事件 E={-7}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"7/12","explanation_zh":"P(S∖E)=7/12。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037079,"parameters":{"S":[-9,-8,-7],"F":"powerset(S)","weights":[3,4,5],"denominator":12,"operation":"complement","E":[-7],"semantic_template":"qf-probability-complement/{\"E\":[-7],\"F\":\"powerset(S)\",\"S\":[-9,-8,-7],\"denominator\":\"12\",\"operation\":\"complement\",\"weights\":[[-9,\"3\"],[-8,\"4\"],[-7,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0080","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-8, -7, -6}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/11, 5/11, 6/11]，事件 E={-8, -7}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"6/11","explanation_zh":"P(S∖E)=6/11。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037080,"parameters":{"S":[-8,-7,-6],"F":"powerset(S)","weights":[0,5,6],"denominator":11,"operation":"complement","E":[-8,-7],"semantic_template":"qf-probability-complement/{\"E\":[-8,-7],\"F\":\"powerset(S)\",\"S\":[-8,-7,-6],\"denominator\":\"11\",\"operation\":\"complement\",\"weights\":[[-8,\"0\"],[-7,\"5\"],[-6,\"6\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0081","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-7, -6, -5}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/9, 1/9, 7/9]，事件 E=∅。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037081,"parameters":{"S":[-7,-6,-5],"F":"powerset(S)","weights":[1,1,7],"denominator":9,"operation":"complement","E":[],"semantic_template":"qf-probability-complement/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-7,-6,-5],\"denominator\":\"9\",\"operation\":\"complement\",\"weights\":[[-7,\"1\"],[-6,\"1\"],[-5,\"7\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0082","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-6, -5, -4}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/12, 2/12, 8/12]，事件 E={-6, -5, -4}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(S∖E)=0/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037082,"parameters":{"S":[-6,-5,-4],"F":"powerset(S)","weights":[2,2,8],"denominator":12,"operation":"complement","E":[-6,-5,-4],"semantic_template":"qf-probability-complement/{\"E\":[-6,-5,-4],\"F\":\"powerset(S)\",\"S\":[-6,-5,-4],\"denominator\":\"6\",\"operation\":\"complement\",\"weights\":[[-6,\"1\"],[-5,\"1\"],[-4,\"4\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0083","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-5, -4, -3}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/15, 3/15, 9/15]，事件 E={-5}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"4/5","explanation_zh":"P(S∖E)=4/5。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037083,"parameters":{"S":[-5,-4,-3],"F":"powerset(S)","weights":[3,3,9],"denominator":15,"operation":"complement","E":[-5],"semantic_template":"qf-probability-complement/{\"E\":[-5],\"F\":\"powerset(S)\",\"S\":[-5,-4,-3],\"denominator\":\"5\",\"operation\":\"complement\",\"weights\":[[-5,\"1\"],[-4,\"1\"],[-3,\"3\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0084","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-4, -3, -2}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/14, 4/14, 10/14]，事件 E={-2}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"2/7","explanation_zh":"P(S∖E)=2/7。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037084,"parameters":{"S":[-4,-3,-2],"F":"powerset(S)","weights":[0,4,10],"denominator":14,"operation":"complement","E":[-2],"semantic_template":"qf-probability-complement/{\"E\":[-2],\"F\":\"powerset(S)\",\"S\":[-4,-3,-2],\"denominator\":\"7\",\"operation\":\"complement\",\"weights\":[[-4,\"0\"],[-3,\"2\"],[-2,\"5\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0085","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-3, -2, -1}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/17, 5/17, 11/17]，事件 E={-3, -2}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"11/17","explanation_zh":"P(S∖E)=11/17。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037085,"parameters":{"S":[-3,-2,-1],"F":"powerset(S)","weights":[1,5,11],"denominator":17,"operation":"complement","E":[-3,-2],"semantic_template":"qf-probability-complement/{\"E\":[-3,-2],\"F\":\"powerset(S)\",\"S\":[-3,-2,-1],\"denominator\":\"17\",\"operation\":\"complement\",\"weights\":[[-3,\"1\"],[-2,\"5\"],[-1,\"11\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0086","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-2, -1, 0}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/15, 1/15, 12/15]，事件 E=∅。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037086,"parameters":{"S":[-2,-1,0],"F":"powerset(S)","weights":[2,1,12],"denominator":15,"operation":"complement","E":[],"semantic_template":"qf-probability-complement/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[-2,-1,0],\"denominator\":\"15\",\"operation\":\"complement\",\"weights\":[[-2,\"2\"],[-1,\"1\"],[0,\"12\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0087","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={-1, 0, 1}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/18, 2/18, 13/18]，事件 E={-1, 0, 1}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(S∖E)=0/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037087,"parameters":{"S":[-1,0,1],"F":"powerset(S)","weights":[3,2,13],"denominator":18,"operation":"complement","E":[-1,0,1],"semantic_template":"qf-probability-complement/{\"E\":[-1,0,1],\"F\":\"powerset(S)\",\"S\":[-1,0,1],\"denominator\":\"18\",\"operation\":\"complement\",\"weights\":[[-1,\"3\"],[0,\"2\"],[1,\"13\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0088","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={0, 1, 2}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/17, 3/17, 14/17]，事件 E={0}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037088,"parameters":{"S":[0,1,2],"F":"powerset(S)","weights":[0,3,14],"denominator":17,"operation":"complement","E":[0],"semantic_template":"qf-probability-complement/{\"E\":[0],\"F\":\"powerset(S)\",\"S\":[0,1,2],\"denominator\":\"17\",\"operation\":\"complement\",\"weights\":[[0,\"0\"],[1,\"3\"],[2,\"14\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0089","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={1, 2, 3}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/20, 4/20, 15/20]，事件 E={3}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/4","explanation_zh":"P(S∖E)=1/4。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037089,"parameters":{"S":[1,2,3],"F":"powerset(S)","weights":[1,4,15],"denominator":20,"operation":"complement","E":[3],"semantic_template":"qf-probability-complement/{\"E\":[3],\"F\":\"powerset(S)\",\"S\":[1,2,3],\"denominator\":\"20\",\"operation\":\"complement\",\"weights\":[[1,\"1\"],[2,\"4\"],[3,\"15\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0090","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={2, 3, 4}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/23, 5/23, 16/23]，事件 E={2, 3}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"16/23","explanation_zh":"P(S∖E)=16/23。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037090,"parameters":{"S":[2,3,4],"F":"powerset(S)","weights":[2,5,16],"denominator":23,"operation":"complement","E":[2,3],"semantic_template":"qf-probability-complement/{\"E\":[2,3],\"F\":\"powerset(S)\",\"S\":[2,3,4],\"denominator\":\"23\",\"operation\":\"complement\",\"weights\":[[2,\"2\"],[3,\"5\"],[4,\"16\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0091","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={3, 4, 5}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/21, 1/21, 17/21]，事件 E=∅。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037091,"parameters":{"S":[3,4,5],"F":"powerset(S)","weights":[3,1,17],"denominator":21,"operation":"complement","E":[],"semantic_template":"qf-probability-complement/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[3,4,5],\"denominator\":\"21\",\"operation\":\"complement\",\"weights\":[[3,\"3\"],[4,\"1\"],[5,\"17\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0092","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={4, 5, 6}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/20, 2/20, 18/20]，事件 E={4, 5, 6}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(S∖E)=0/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037092,"parameters":{"S":[4,5,6],"F":"powerset(S)","weights":[0,2,18],"denominator":20,"operation":"complement","E":[4,5,6],"semantic_template":"qf-probability-complement/{\"E\":[4,5,6],\"F\":\"powerset(S)\",\"S\":[4,5,6],\"denominator\":\"10\",\"operation\":\"complement\",\"weights\":[[4,\"0\"],[5,\"1\"],[6,\"9\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0093","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={5, 6, 7}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/23, 3/23, 19/23]，事件 E={5}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"22/23","explanation_zh":"P(S∖E)=22/23。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037093,"parameters":{"S":[5,6,7],"F":"powerset(S)","weights":[1,3,19],"denominator":23,"operation":"complement","E":[5],"semantic_template":"qf-probability-complement/{\"E\":[5],\"F\":\"powerset(S)\",\"S\":[5,6,7],\"denominator\":\"23\",\"operation\":\"complement\",\"weights\":[[5,\"1\"],[6,\"3\"],[7,\"19\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0094","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={6, 7, 8}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/26, 4/26, 20/26]，事件 E={8}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"3/13","explanation_zh":"P(S∖E)=3/13。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037094,"parameters":{"S":[6,7,8],"F":"powerset(S)","weights":[2,4,20],"denominator":26,"operation":"complement","E":[8],"semantic_template":"qf-probability-complement/{\"E\":[8],\"F\":\"powerset(S)\",\"S\":[6,7,8],\"denominator\":\"13\",\"operation\":\"complement\",\"weights\":[[6,\"1\"],[7,\"2\"],[8,\"10\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0095","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={7, 8, 9}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/29, 5/29, 21/29]，事件 E={7, 8}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"21/29","explanation_zh":"P(S∖E)=21/29。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037095,"parameters":{"S":[7,8,9],"F":"powerset(S)","weights":[3,5,21],"denominator":29,"operation":"complement","E":[7,8],"semantic_template":"qf-probability-complement/{\"E\":[7,8],\"F\":\"powerset(S)\",\"S\":[7,8,9],\"denominator\":\"29\",\"operation\":\"complement\",\"weights\":[[7,\"3\"],[8,\"5\"],[9,\"21\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0096","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={8, 9, 10}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/23, 1/23, 22/23]，事件 E=∅。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"1/1","explanation_zh":"P(S∖E)=1/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037096,"parameters":{"S":[8,9,10],"F":"powerset(S)","weights":[0,1,22],"denominator":23,"operation":"complement","E":[],"semantic_template":"qf-probability-complement/{\"E\":[],\"F\":\"powerset(S)\",\"S\":[8,9,10],\"denominator\":\"23\",\"operation\":\"complement\",\"weights\":[[8,\"0\"],[9,\"1\"],[10,\"22\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0097","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={9, 10, 11}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [1/26, 2/26, 23/26]，事件 E={9, 10, 11}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(S∖E)=0/1。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037097,"parameters":{"S":[9,10,11],"F":"powerset(S)","weights":[1,2,23],"denominator":26,"operation":"complement","E":[9,10,11],"semantic_template":"qf-probability-complement/{\"E\":[9,10,11],\"F\":\"powerset(S)\",\"S\":[9,10,11],\"denominator\":\"26\",\"operation\":\"complement\",\"weights\":[[9,\"1\"],[10,\"2\"],[11,\"23\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0098","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={10, 11, 12}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [2/29, 3/29, 24/29]，事件 E={10}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"27/29","explanation_zh":"P(S∖E)=27/29。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037098,"parameters":{"S":[10,11,12],"F":"powerset(S)","weights":[2,3,24],"denominator":29,"operation":"complement","E":[10],"semantic_template":"qf-probability-complement/{\"E\":[10],\"F\":\"powerset(S)\",\"S\":[10,11,12],\"denominator\":\"29\",\"operation\":\"complement\",\"weights\":[[10,\"2\"],[11,\"3\"],[12,\"24\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0099","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={11, 12, 13}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [3/32, 4/32, 25/32]，事件 E={13}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"7/32","explanation_zh":"P(S∖E)=7/32。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037099,"parameters":{"S":[11,12,13],"F":"powerset(S)","weights":[3,4,25],"denominator":32,"operation":"complement","E":[13],"semantic_template":"qf-probability-complement/{\"E\":[13],\"F\":\"powerset(S)\",\"S\":[11,12,13],\"denominator\":\"32\",\"operation\":\"complement\",\"weights\":[[11,\"3\"],[12,\"4\"],[13,\"25\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-r07-probability-0100","batch_id":"batch-20261003-r07-probability-foundations","language":"zh-Hant","source_mko_ids":["mko-probability"],"title_zh":"加權模型的補事件機率","stem_zh":"非空有限樣本空間 S={12, 13, 14}，事件族 F=𝒫(S)。依 S 列示順序的權重為 [0/31, 5/31, 26/31]，事件 E={12, 13}。求精確 P(S∖E)，不可假設等可能。","answer":{"kind":"expression","value":"26/31","explanation_zh":"P(S∖E)=26/31。先驗證非負權重總和為一，再相加補事件成員權重；非空事件也能機率為零。𝒫 是全冪集記號，P 是機率律，兩者不同。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["加權模型的補事件機率"]},"variant":{"family_id":"qf-probability-complement","seed":202610037100,"parameters":{"S":[12,13,14],"F":"powerset(S)","weights":[0,5,26],"denominator":31,"operation":"complement","E":[12,13],"semantic_template":"qf-probability-complement/{\"E\":[12,13],\"F\":\"powerset(S)\",\"S\":[12,13,14],\"denominator\":\"31\",\"operation\":\"complement\",\"weights\":[[12,\"0\"],[13,\"5\"],[14,\"26\"]]}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為獨立出題上下文的候選與自檢；主架構者仍須獨立驗收，不構成人工審定或一般形式證明。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}