{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261030-r07-convergence-in-probability-0023", "pilot_number": "R0029-15023", "number": 15023, "family_id": "qf-mod30-convprob-deviation-probability", "title": { "zh-Hant": "第 15023 題：依機率收斂", "en": "Question 15023: Evaluate the Deviation Profile Only at a Valid Probability Index" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261030-r07-convergence-in-probability-0023", "batch_id": "batch-20261014-advance-20261030-r07-convergence-in-probability-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-convergence-in-probability" ], "title_zh": "依機率收斂", "stem_zh": "共同機率空間與forall epsilon tail責任。操作=convprob-deviation-probability；資料={\"numerator\":3,\"shift\":3,\"n\":23}。請依 [numerator,shift,n,probability] 欄序回答。", "answer": { "kind": "expression", "value": "(3, 3, 23, 3/26)", "explanation_zh": "Wave4第四層核對=(3, 3, 23, 3/26)；依完整明示資料重算；完整有限／symbolic tail證書不自動升格為未聲明的一般定理；finite prefix不足且只接受symbolic tail或eventual equality；不混同almost sure Lp或distribution convergence。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "依機率收斂" ] }, "variant": { "family_id": "qf-mod30-convprob-deviation-probability", "seed": 202610307023, "parameters": { "numerator": 3, "shift": 3, "n": 23, "domain": "Symbolic_deviation_probability_tail", "operation": "convprob-deviation-probability", "semantic_template": "qf-mod30-convprob-deviation-probability/{\"n\":23,\"numerator\":3,\"shift\":3}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "候選須主端重算；不構成人工審定、一般域／Hilbert／martingale／無限基數／一般收斂、Lean或Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261030-r07-convergence-in-probability-0023", "batch_id": "batch-20261014-advance-20261030-r07-convergence-in-probability-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "4905707f3edfef4fb8c860639222d253e9426cdc6b84b71c748e14f55d44f156", "family_id": "qf-mod30-convprob-deviation-probability", "seed": 202610307023, "semantic_template": "qf-mod30-convprob-deviation-probability/{\"n\":23,\"numerator\":3,\"shift\":3}" }, "fields": { "title": "Evaluate the Deviation Profile Only at a Valid Probability Index", "stem": "Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=23. Give (numerator,shift,n,deviation_probability).", "answer": { "kind": "expression", "value": "(3, 3, 23, 3/26)", "explanation": "The supplied index must lie in the declared probability domain. Evaluate the exact rational value without clipping it to1, renormalizing it or silently shifting the index. One value does not establish convergence for every epsilon. The exact result, retaining the requested order, is (3, 3, 23, 3/26)." }, "skills": [ "Evaluate the Deviation Profile Only at a Valid Probability Index" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "共同機率空間與forall epsilon tail責任。操作=convprob-deviation-probability；資料={\"numerator\":3,\"shift\":3,\"n\":23}。請依 [numerator,shift,n,probability] 欄序回答。", "en": "Use the declared rational deviation-probability tail p_n=3/(n+3) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=23. Give (numerator,shift,n,deviation_probability)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "positive profile parameters c=3;s=3;raw fraction=c/(n+s)", "after": "nonnegative valid probability indices start at=0;only there0