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R0055-28347 · q-20261003-r07-probability-0072
The nonempty finite sample space is S={9, 10, 11} and the event family is F=𝒫(S). In the displayed order of S, the proposed weights are [-22/(24), 2/(24), 0/(24)]. Do they define a valid probability model with nonnegative weights summing to one?
G001 — The nonempty finite sample space is S={9, 10, 11} and the event family is F=𝒫(S). In the displayed order of S, the proposed weights are [-22/(24), 2/(24), 0/(24)]. Do they define a valid probability model with nonnegative weights summing to one?Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
source roles = [qf-probability-model-validity, model_validity]
After
derived = [[candidate_probability_model_valid_yes_no]]
Conditions: none
List this scalar/label/set/point answer role or mode-specific ordered fields,not an imposed universal tuple.Retain legacy domain/underscore operation and integer fractions with/1;original bilingual snapshots stay unchanged.
Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001
Before
source roles = [[9, 10, 11], powerset(S)]
After
derived = [[9, 10, 11], 3]
Conditions: none
S is a nonempty set of at most100 distinct SAFE integers and F is explicitly the FULL powerset,not an event family inferred from one event.
Rule: Legacy_probability_NONEMPTY_S_FULLpowerset_exact_declared_event_family · Uses: G001, S001
Before
Legacy probability ORIGINAL ALL numerator weights sum := sum(-22, 2, 0)
After
Legacy probability ORIGINAL ALL numerator weights sum = -20
Conditions: none
Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [[9, 10, 11], [-22, 2, 0], 24]
After
derived = [[[9, -22], [10, 2], [11, 0]], -20, Yes, No, No, No]
Conditions: none
Bind weights to ORIGINAL raw S slots;set sorting does not change these associations.Validity needs a positive denominator,all nonnegative weights and the BigInt total equal to the denominator.A model-validity question may legally answerNo,without0/0,sign repair or renormalizing another model.
Rule: Legacy_probability_ORIGINAL_rawS_weight_alignment_and_model_qualification · Uses: G001, S003
S004 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.S003 · premise_and_result_audit — Recompute Legacy probability ORIGINAL ALL numerator weights sum with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.S001 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.No
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 8ee457277c9fc202363e53ae39368b0ce5e07401efd40a137c5aa7efe1340c60candidate record SHA-256: 23d0d73eae687b1e1cd48631cc8944af3370fa651d823a1d0320f6bf5316c263semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4eLean: NOT_RUN · human review: NOT_PERFORMED