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Question 28335: Validity of a Candidate Finite Probability Model

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0055-28335 · q-20261003-r07-probability-0060

Original question and givens

The nonempty finite sample space is S={-3, -2, -1} and the event family is F=𝒫(S). In the displayed order of S, the proposed weights are [-10/(15), 5/(15), 0/(15)]. Do they define a valid probability model with nonnegative weights summing to one?

  • G001 — The nonempty finite sample space is S={-3, -2, -1} and the event family is F=𝒫(S). In the displayed order of S, the proposed weights are [-10/(15), 5/(15), 0/(15)]. Do they define a valid probability model with nonnegative weights summing to one?

Definition and scope

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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[-3, -2, -1], powerset(S)]

After

derived = [[-3, -2, -1], 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.

Step identifiers and dependencies

Rule: Legacy_probability_NONEMPTY_S_FULLpowerset_exact_declared_event_family · Uses: G001, S001

S003 · calculation

Before

Legacy probability ORIGINAL ALL numerator weights sum := sum(-10, 5, 0)

After

Legacy probability ORIGINAL ALL numerator weights sum = -5

Conditions: none

Substitute original roles exactly,without floating approximation or changed operands;extra teaching identities add no source-domain restrictions.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [[-3, -2, -1], [-10, 5, 0], 15]

After

derived = [[[-3, -10], [-2, 5], [-1, 0]], -5, 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.

Step identifiers and dependencies

Rule: Legacy_probability_ORIGINAL_rawS_weight_alignment_and_model_qualification · Uses: G001, S003

Reverse checks

  • 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.

Result

No

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 5fc6b4c81a52c1c1835842fd54f67283bb681a75882651256e6d81eaa57b01a2
  • candidate record SHA-256: ed5602498181eb6da03d853097fb4161f1cf5de359d57b72858080a09c0b330a
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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