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Question 28325: Complement Probability in a Weighted 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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R0055-28325 · q-20261003-r07-probability-0100

Original question and givens

The nonempty finite sample space is S={12, 13, 14} and the event family is F=𝒫(S). In the displayed order of S, the weights are [0/(31), 5/(31), 26/(31)] and E={12, 13}. Find P(S∖E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.

  • G001 — The nonempty finite sample space is S={12, 13, 14} and the event family is F=𝒫(S). In the displayed order of S, the weights are [0/(31), 5/(31), 26/(31)] and E={12, 13}. Find P(S∖E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.

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-complement, complement]

After

derived = [[reduced_same_carrier_complement_fraction_retaining1]]

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 = [[12, 13, 14], powerset(S)]

After

derived = [[12, 13, 14], 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(0, 5, 26)

After

Legacy probability ORIGINAL ALL numerator weights sum = 31

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 = [[12, 13, 14], [0, 5, 26], 31]

After

derived = [[[12, 0], [13, 5], [14, 26]], 31, Yes, Yes, Yes, Yes]

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

S005 · calculation

Before

Legacy probability ORIGINAL included outcome12 running sum0 := sum(0, 0)

After

Legacy probability ORIGINAL included outcome12 running sum0 = 0

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, S004

S006 · calculation

Before

Legacy probability ORIGINAL included outcome13 running sum1 := sum(0, 5)

After

Legacy probability ORIGINAL included outcome13 running sum1 = 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, S005

S007 · calculation

Before

Legacy complement ORIGINAL denominator minus event mass := sub(31, 5)

After

Legacy complement ORIGINAL denominator minus event mass = 26

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, S006

S008 · calculation

Before

Legacy probability ORIGINAL normalized event or complement fraction := div(26, 31)

After

Legacy probability ORIGINAL normalized event or complement fraction = 26/31

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, S007

S009 · implies

Before

source roles = [[12, 13, 14], [12, 13], [[12, 0], [13, 5], [14, 26]], 31]

After

derived = [5, [14], 26, 26/31]

Conditions: none

On the full powerset define every event by its ORIGINAL weighted sum.Nonnegativity,total1 and regrouping disjoint events give finite(and countable,for finite S)additivity.A nonempty E can have mass0.The complement is in the SAME S;do not substitute sorted-slot weights or uniform counts.

Step identifiers and dependencies

Rule: Legacy_complete_finite_probability_axioms_and_same_carrier_complement · Uses: G001, S008

Reverse checks

  • S009 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.
  • S008 · premise_and_result_audit — Recompute Legacy probability ORIGINAL normalized event or complement fraction with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Legacy complement ORIGINAL denominator minus event mass with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome13 running sum1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome12 running sum0 with exact rational arithmetic, not rounded decimals.
  • 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

26/31

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: 81446b406ed68400873ca23411c1fdd86944f426447ce657cf957be16efd1b8c
  • candidate record SHA-256: 504ac5e595ad73738137e0142ff383aa56f8e55e9aed9c0fd608bc47a40a39ff
  • semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683
  • question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4e
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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