{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0001","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"0","explanation_zh":"X(0)=0。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055001,"parameters":{"S":[0],"F":"powerset(S)","values":[0],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0002","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -1), (1, -1), (2, 3), (3, 1)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-1","explanation_zh":"X(1)=-1。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055002,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-1,-1,3,1],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0003","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 4), (3, 0)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"4","explanation_zh":"X(2)=4。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055003,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,4,0],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0004","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -3), (1, -3), (2, 5), (3, 5)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"5","explanation_zh":"X(3)=5。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055004,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-3,-3,5,5],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0005","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 6), (3, -2)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-4","explanation_zh":"X(0)=-4。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055005,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,6,-2],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0006","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -5), (1, -5), (2, 7), (3, -3)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-5","explanation_zh":"X(1)=-5。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055006,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-5,-5,7,-3],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0007","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 8), (3, 8)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"8","explanation_zh":"X(2)=8。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055007,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,8,8],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0008","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, 9), (3, -5)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-5","explanation_zh":"X(3)=-5。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055008,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,9,-5],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0009","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, 10), (3, -6)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-8","explanation_zh":"X(0)=-8。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055009,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,10,-6],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0010","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, 11), (3, 11)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-9","explanation_zh":"X(1)=-9。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055010,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,11,11],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0011","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -10), (1, -10), (2, 12), (3, -8)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"12","explanation_zh":"X(2)=12。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055011,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,12,-8],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0012","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, 13), (3, -9)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-9","explanation_zh":"X(3)=-9。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055012,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,13,-9],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0013","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -12), (1, -12), (2, 14), (3, 14)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-12","explanation_zh":"X(0)=-12。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055013,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-12,-12,14,14],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0014","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, 15), (3, -11)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-13","explanation_zh":"X(1)=-13。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055014,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,15,-11],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0015","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -14), (1, -14), (2, 16), (3, -12)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"16","explanation_zh":"X(2)=16。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055015,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-14,-14,16,-12],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0016","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, 17), (3, 17)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"17","explanation_zh":"X(3)=17。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055016,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,17,17],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0017","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -16), (1, -16), (2, 18), (3, -14)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-16","explanation_zh":"X(0)=-16。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055017,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-16,-16,18,-14],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0018","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 19), (3, -15)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-17","explanation_zh":"X(1)=-17。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055018,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,19,-15],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0019","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -18), (1, -18), (2, 20), (3, 20)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"20","explanation_zh":"X(2)=20。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055019,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-18,-18,20,20],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0020","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, 21), (3, -17)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-17","explanation_zh":"X(3)=-17。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055020,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,21,-17],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0021","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 22), (3, -18)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-20","explanation_zh":"X(0)=-20。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055021,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-20,-20,22,-18],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0022","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, 23), (3, 23)}。求 X(1)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-21","explanation_zh":"X(1)=-21。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055022,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,23,23],"s":1,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_evaluate\",\"s\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0023","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -22), (1, -22), (2, 24), (3, -20)}。求 X(2)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"24","explanation_zh":"X(2)=24。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055023,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-22,-22,24,-20],"s":2,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_evaluate\",\"s\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0024","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, 25), (3, -21)}。求 X(3)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-21","explanation_zh":"X(3)=-21。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055024,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,25,-21],"s":3,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_evaluate\",\"s\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0025","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"基本結果的隨機變數值","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -24), (1, -24), (2, 26), (3, 26)}。求 X(0)，只讀取函數值，不指派任何未給的機率。","answer":{"kind":"short_text","value":"-24","explanation_zh":"X(0)=-24。先確認基本結果屬於 S，再讀取它的唯一實數值；沒有給機率律時不假設等可能。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["基本結果的隨機變數值"]},"variant":{"family_id":"qf-random-variable-evaluation","seed":202610055025,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-24,-24,26,26],"s":0,"operation":"rv_evaluate","semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_evaluate\",\"s\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0026","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0)}。求事件 {s∈S:X(s)≥-1}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0}","explanation_zh":"事件={0}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055026,"parameters":{"S":[0],"F":"powerset(S)","values":[0],"t":-1,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_threshold\",\"t\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0027","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -1), (1, -1), (2, 3), (3, 1)}。求事件 {s∈S:X(s)≥4}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055027,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-1,-1,3,1],"t":4,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_threshold\",\"t\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0028","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 4), (3, 0)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2, 3}","explanation_zh":"事件={2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055028,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,4,0],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0029","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -3), (1, -3), (2, 5), (3, 5)}。求事件 {s∈S:X(s)≥-3}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055029,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-3,-3,5,5],"t":-3,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_threshold\",\"t\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0030","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 6), (3, -2)}。求事件 {s∈S:X(s)≥-5}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055030,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,6,-2],"t":-5,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_threshold\",\"t\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0031","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -5), (1, -5), (2, 7), (3, -3)}。求事件 {s∈S:X(s)≥8}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055031,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-5,-5,7,-3],"t":8,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_threshold\",\"t\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0032","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 8), (3, 8)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2, 3}","explanation_zh":"事件={2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055032,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,8,8],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0033","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, 9), (3, -5)}。求事件 {s∈S:X(s)≥-7}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055033,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,9,-5],"t":-7,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_threshold\",\"t\":-7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0034","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, 10), (3, -6)}。求事件 {s∈S:X(s)≥-9}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055034,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,10,-6],"t":-9,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_threshold\",\"t\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0035","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, 11), (3, 11)}。求事件 {s∈S:X(s)≥12}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055035,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,11,11],"t":12,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_threshold\",\"t\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0036","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -10), (1, -10), (2, 12), (3, -8)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2}","explanation_zh":"事件={2}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055036,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,12,-8],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0037","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, 13), (3, -9)}。求事件 {s∈S:X(s)≥-11}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055037,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,13,-9],"t":-11,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_threshold\",\"t\":-11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0038","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -12), (1, -12), (2, 14), (3, 14)}。求事件 {s∈S:X(s)≥-13}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055038,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-12,-12,14,14],"t":-13,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_threshold\",\"t\":-13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0039","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, 15), (3, -11)}。求事件 {s∈S:X(s)≥16}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055039,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,15,-11],"t":16,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_threshold\",\"t\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0040","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -14), (1, -14), (2, 16), (3, -12)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2}","explanation_zh":"事件={2}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055040,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-14,-14,16,-12],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0041","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, 17), (3, 17)}。求事件 {s∈S:X(s)≥-15}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055041,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,17,17],"t":-15,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_threshold\",\"t\":-15}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0042","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -16), (1, -16), (2, 18), (3, -14)}。求事件 {s∈S:X(s)≥-17}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055042,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-16,-16,18,-14],"t":-17,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_threshold\",\"t\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0043","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 19), (3, -15)}。求事件 {s∈S:X(s)≥20}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055043,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,19,-15],"t":20,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_threshold\",\"t\":20}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0044","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -18), (1, -18), (2, 20), (3, 20)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2, 3}","explanation_zh":"事件={2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055044,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-18,-18,20,20],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0045","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, 21), (3, -17)}。求事件 {s∈S:X(s)≥-19}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055045,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,21,-17],"t":-19,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_threshold\",\"t\":-19}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0046","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 22), (3, -18)}。求事件 {s∈S:X(s)≥-21}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055046,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-20,-20,22,-18],"t":-21,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_threshold\",\"t\":-21}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0047","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, 23), (3, 23)}。求事件 {s∈S:X(s)≥24}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"∅","explanation_zh":"事件=∅。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055047,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,23,23],"t":24,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_threshold\",\"t\":24}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0048","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -22), (1, -22), (2, 24), (3, -20)}。求事件 {s∈S:X(s)≥0}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{2}","explanation_zh":"事件={2}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055048,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-22,-22,24,-20],"t":0,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_threshold\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0049","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, 25), (3, -21)}。求事件 {s∈S:X(s)≥-23}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055049,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,25,-21],"t":-23,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_threshold\",\"t\":-23}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0050","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"閾值條件的原像事件","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -24), (1, -24), (2, 26), (3, 26)}。求事件 {s∈S:X(s)≥-25}，答基本結果集合而不是機率數值。","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation_zh":"事件={0, 1, 2, 3}。逐結果檢查含等號的閾值條件；F 是全冪集，得到的子集合為事件，未給權重不宣稱機率。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["閾值條件的原像事件"]},"variant":{"family_id":"qf-random-variable-threshold-event","seed":202610055050,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-24,-24,26,26],"t":-25,"operation":"rv_threshold","semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_threshold\",\"t\":-25}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0051","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{0}","explanation_zh":"值域={0}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055051,"parameters":{"S":[0],"F":"powerset(S)","values":[0],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0052","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -1), (1, -1), (2, 3), (3, 1)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-1, 1, 3}","explanation_zh":"值域={-1, 1, 3}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055052,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-1,-1,3,1],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0053","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 4), (3, 0)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-2, 0, 4}","explanation_zh":"值域={-2, 0, 4}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055053,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,4,0],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0054","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -3), (1, -3), (2, 5), (3, 5)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-3, 5}","explanation_zh":"值域={-3, 5}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055054,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-3,-3,5,5],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0055","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 6), (3, -2)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-4, -2, 6}","explanation_zh":"值域={-4, -2, 6}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055055,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,6,-2],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0056","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -5), (1, -5), (2, 7), (3, -3)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-5, -3, 7}","explanation_zh":"值域={-5, -3, 7}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055056,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-5,-5,7,-3],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0057","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 8), (3, 8)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-6, 8}","explanation_zh":"值域={-6, 8}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055057,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,8,8],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0058","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, 9), (3, -5)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-7, -5, 9}","explanation_zh":"值域={-7, -5, 9}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055058,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,9,-5],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0059","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, 10), (3, -6)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-8, -6, 10}","explanation_zh":"值域={-8, -6, 10}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055059,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,10,-6],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0060","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, 11), (3, 11)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-9, 11}","explanation_zh":"值域={-9, 11}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055060,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,11,11],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0061","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -10), (1, -10), (2, 12), (3, -8)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-10, -8, 12}","explanation_zh":"值域={-10, -8, 12}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055061,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,12,-8],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0062","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, 13), (3, -9)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-11, -9, 13}","explanation_zh":"值域={-11, -9, 13}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055062,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,13,-9],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0063","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -12), (1, -12), (2, 14), (3, 14)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-12, 14}","explanation_zh":"值域={-12, 14}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055063,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-12,-12,14,14],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0064","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, 15), (3, -11)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-13, -11, 15}","explanation_zh":"值域={-13, -11, 15}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055064,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,15,-11],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0065","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -14), (1, -14), (2, 16), (3, -12)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-14, -12, 16}","explanation_zh":"值域={-14, -12, 16}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055065,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-14,-14,16,-12],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0066","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, 17), (3, 17)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-15, 17}","explanation_zh":"值域={-15, 17}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055066,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,17,17],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0067","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -16), (1, -16), (2, 18), (3, -14)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-16, -14, 18}","explanation_zh":"值域={-16, -14, 18}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055067,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-16,-16,18,-14],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0068","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 19), (3, -15)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-17, -15, 19}","explanation_zh":"值域={-17, -15, 19}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055068,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,19,-15],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0069","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -18), (1, -18), (2, 20), (3, 20)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-18, 20}","explanation_zh":"值域={-18, 20}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055069,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-18,-18,20,20],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0070","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, 21), (3, -17)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-19, -17, 21}","explanation_zh":"值域={-19, -17, 21}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055070,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,21,-17],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0071","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 22), (3, -18)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-20, -18, 22}","explanation_zh":"值域={-20, -18, 22}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055071,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-20,-20,22,-18],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0072","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, 23), (3, 23)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-21, 23}","explanation_zh":"值域={-21, 23}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055072,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,23,23],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0073","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -22), (1, -22), (2, 24), (3, -20)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-22, -20, 24}","explanation_zh":"值域={-22, -20, 24}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055073,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-22,-22,24,-20],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0074","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, 25), (3, -21)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-23, -21, 25}","explanation_zh":"值域={-23, -21, 25}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055074,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,25,-21],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0075","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"隨機變數的實際值域","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -24), (1, -24), (2, 26), (3, 26)}。求 X[S] 的集合，重複輸出只列一次。","answer":{"kind":"short_text","value":"{-24, 26}","explanation_zh":"值域={-24, 26}。值域是實際輸出而不是基本結果或機率權重；去掉重複值不等於把原像權重丟掉。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數的實際值域"]},"variant":{"family_id":"qf-random-variable-range","seed":202610055075,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-24,-24,26,26],"operation":"rv_range","semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_range\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0076","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -1), (1, -1), (2, 2), (3, 999)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/6, 2/6, 3/6, 0/6]，權重總和恰為一。求精確 P(X=-1)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"1/2","explanation_zh":"P(X=-1)=1/2。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055076,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-1,-1,2,999],"weights":[1,2,3,0],"denominator":6,"t":-1,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-1\",\"1/2\"],[\"2\",\"1/2\"]],\"operation\":\"rv_point_mass\",\"t\":-1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0077","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 3), (3, 1000)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/7, 3/7, 3/7, 0/7]，權重總和恰為一。求精確 P(X=3)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"3/7","explanation_zh":"P(X=3)=3/7。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055077,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,3,1000],"weights":[1,3,3,0],"denominator":7,"t":3,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-2\",\"4/7\"],[\"3\",\"3/7\"]],\"operation\":\"rv_point_mass\",\"t\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0078","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -3), (1, -3), (2, 4), (3, 1001)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/8, 4/8, 3/8, 0/8]，權重總和恰為一。求精確 P(X=1001)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1001)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055078,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-3,-3,4,1001],"weights":[1,4,3,0],"denominator":8,"t":1001,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-3\",\"5/8\"],[\"4\",\"3/8\"]],\"operation\":\"rv_point_mass\",\"t\":1001}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0079","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 5), (3, 1002)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/9, 5/9, 3/9, 0/9]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055079,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,5,1002],"weights":[1,5,3,0],"denominator":9,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-4\",\"2/3\"],[\"5\",\"1/3\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0080","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -5), (1, -5), (2, 6), (3, 1003)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 6/10, 3/10, 0/10]，權重總和恰為一。求精確 P(X=-5)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"7/10","explanation_zh":"P(X=-5)=7/10。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055080,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-5,-5,6,1003],"weights":[1,6,3,0],"denominator":10,"t":-5,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-5\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"rv_point_mass\",\"t\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0081","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 7), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/11, 7/11, 3/11, 0/11]，權重總和恰為一。求精確 P(X=7)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"3/11","explanation_zh":"P(X=7)=3/11。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055081,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,7,1004],"weights":[1,7,3,0],"denominator":11,"t":7,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-6\",\"8/11\"],[\"7\",\"3/11\"]],\"operation\":\"rv_point_mass\",\"t\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0082","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, 8), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 8/12, 3/12, 0/12]，權重總和恰為一。求精確 P(X=1005)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1005)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055082,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,8,1005],"weights":[1,8,3,0],"denominator":12,"t":1005,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-7\",\"3/4\"],[\"8\",\"1/4\"]],\"operation\":\"rv_point_mass\",\"t\":1005}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0083","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, 9), (3, 1006)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/13, 9/13, 3/13, 0/13]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055083,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,9,1006],"weights":[1,9,3,0],"denominator":13,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-8\",\"10/13\"],[\"9\",\"3/13\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0084","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, 10), (3, 1007)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 10/14, 3/14, 0/14]，權重總和恰為一。求精確 P(X=-9)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"11/14","explanation_zh":"P(X=-9)=11/14。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055084,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,10,1007],"weights":[1,10,3,0],"denominator":14,"t":-9,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-9\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"rv_point_mass\",\"t\":-9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0085","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -10), (1, -10), (2, 11), (3, 1008)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/15, 11/15, 3/15, 0/15]，權重總和恰為一。求精確 P(X=11)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"1/5","explanation_zh":"P(X=11)=1/5。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055085,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,11,1008],"weights":[1,11,3,0],"denominator":15,"t":11,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-10\",\"4/5\"],[\"11\",\"1/5\"]],\"operation\":\"rv_point_mass\",\"t\":11}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0086","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, 12), (3, 1009)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 12/16, 3/16, 0/16]，權重總和恰為一。求精確 P(X=1009)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1009)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055086,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,12,1009],"weights":[1,12,3,0],"denominator":16,"t":1009,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-11\",\"13/16\"],[\"12\",\"3/16\"]],\"operation\":\"rv_point_mass\",\"t\":1009}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0087","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -12), (1, -12), (2, 13), (3, 1010)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/17, 13/17, 3/17, 0/17]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055087,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-12,-12,13,1010],"weights":[1,13,3,0],"denominator":17,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-12\",\"14/17\"],[\"13\",\"3/17\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0088","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, 14), (3, 1011)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 14/18, 3/18, 0/18]，權重總和恰為一。求精確 P(X=-13)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"5/6","explanation_zh":"P(X=-13)=5/6。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055088,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,14,1011],"weights":[1,14,3,0],"denominator":18,"t":-13,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-13\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"rv_point_mass\",\"t\":-13}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0089","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -14), (1, -14), (2, 15), (3, 1012)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/19, 15/19, 3/19, 0/19]，權重總和恰為一。求精確 P(X=15)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"3/19","explanation_zh":"P(X=15)=3/19。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055089,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-14,-14,15,1012],"weights":[1,15,3,0],"denominator":19,"t":15,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-14\",\"16/19\"],[\"15\",\"3/19\"]],\"operation\":\"rv_point_mass\",\"t\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0090","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, 16), (3, 1013)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 16/20, 3/20, 0/20]，權重總和恰為一。求精確 P(X=1013)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1013)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055090,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,16,1013],"weights":[1,16,3,0],"denominator":20,"t":1013,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-15\",\"17/20\"],[\"16\",\"3/20\"]],\"operation\":\"rv_point_mass\",\"t\":1013}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0091","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -16), (1, -16), (2, 17), (3, 1014)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/21, 17/21, 3/21, 0/21]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055091,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-16,-16,17,1014],"weights":[1,17,3,0],"denominator":21,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-16\",\"6/7\"],[\"17\",\"1/7\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0092","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, 18), (3, 1015)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 18/22, 3/22, 0/22]，權重總和恰為一。求精確 P(X=-17)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"19/22","explanation_zh":"P(X=-17)=19/22。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055092,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,18,1015],"weights":[1,18,3,0],"denominator":22,"t":-17,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-17\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"rv_point_mass\",\"t\":-17}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0093","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -18), (1, -18), (2, 19), (3, 1016)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/23, 19/23, 3/23, 0/23]，權重總和恰為一。求精確 P(X=19)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"3/23","explanation_zh":"P(X=19)=3/23。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055093,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-18,-18,19,1016],"weights":[1,19,3,0],"denominator":23,"t":19,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-18\",\"20/23\"],[\"19\",\"3/23\"]],\"operation\":\"rv_point_mass\",\"t\":19}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0094","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, 20), (3, 1017)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 20/24, 3/24, 0/24]，權重總和恰為一。求精確 P(X=1017)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1017)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055094,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,20,1017],"weights":[1,20,3,0],"denominator":24,"t":1017,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-19\",\"7/8\"],[\"20\",\"1/8\"]],\"operation\":\"rv_point_mass\",\"t\":1017}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0095","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -20), (1, -20), (2, 21), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/25, 21/25, 3/25, 0/25]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055095,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-20,-20,21,1018],"weights":[1,21,3,0],"denominator":25,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-20\",\"22/25\"],[\"21\",\"3/25\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0096","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, 22), (3, 1019)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 22/26, 3/26, 0/26]，權重總和恰為一。求精確 P(X=-21)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"23/26","explanation_zh":"P(X=-21)=23/26。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055096,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,22,1019],"weights":[1,22,3,0],"denominator":26,"t":-21,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-21\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"rv_point_mass\",\"t\":-21}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0097","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -22), (1, -22), (2, 23), (3, 1020)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/27, 23/27, 3/27, 0/27]，權重總和恰為一。求精確 P(X=23)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"1/9","explanation_zh":"P(X=23)=1/9。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055097,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-22,-22,23,1020],"weights":[1,23,3,0],"denominator":27,"t":23,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-22\",\"8/9\"],[\"23\",\"1/9\"]],\"operation\":\"rv_point_mass\",\"t\":23}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0098","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, 24), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 24/28, 3/28, 0/28]，權重總和恰為一。求精確 P(X=1021)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=1021)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055098,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,24,1021],"weights":[1,24,3,0],"denominator":28,"t":1021,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-23\",\"25/28\"],[\"24\",\"3/28\"]],\"operation\":\"rv_point_mass\",\"t\":1021}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0099","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -24), (1, -24), (2, 25), (3, 1022)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/29, 25/29, 3/29, 0/29]，權重總和恰為一。求精確 P(X=0)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"0/1","explanation_zh":"P(X=0)=0/1。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055099,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-24,-24,25,1022],"weights":[1,25,3,0],"denominator":29,"t":0,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-24\",\"26/29\"],[\"25\",\"3/29\"]],\"operation\":\"rv_point_mass\",\"t\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r05-random-variable-0100","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable"],"title_zh":"重複值的有限點機率","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -25), (1, -25), (2, 26), (3, 1023)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 26/30, 3/30, 0/30]，權重總和恰為一。求精確 P(X=-25)，以最簡正分母 p/q 回答。","answer":{"kind":"expression","value":"9/10","explanation_zh":"P(X=-25)=9/10。把所有值等於指定 t 的結果權重相加；零權重原像不增加質量，未命中時為 0/1，不以值域基數作分母。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["重複值的有限點機率"]},"variant":{"family_id":"qf-random-variable-point-mass","seed":202610055100,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,26,1023],"weights":[1,26,3,0],"denominator":30,"t":-25,"operation":"rv_point_mass","semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-25\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"rv_point_mass\",\"t\":-25}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}