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R0055-28312 · q-20261003-r07-probability-0087
The nonempty finite sample space is S={-1, 0, 1} and the event family is F=𝒫(S). In the displayed order of S, the weights are [3/(18), 2/(18), 13/(18)] and E={-1, 0, 1}. Find P(S∖E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.
G001 — The nonempty finite sample space is S={-1, 0, 1} and the event family is F=𝒫(S). In the displayed order of S, the weights are [3/(18), 2/(18), 13/(18)] and E={-1, 0, 1}. Find P(S∖E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.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-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.
Rule: source_answer_roles_or_ordered_fields_without_JS · Uses: G001
Before
source roles = [[-1, 0, 1], powerset(S)]
After
derived = [[-1, 0, 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.
Rule: Legacy_probability_NONEMPTY_S_FULLpowerset_exact_declared_event_family · Uses: G001, S001
Before
Legacy probability ORIGINAL ALL numerator weights sum := sum(3, 2, 13)
After
Legacy probability ORIGINAL ALL numerator weights sum = 18
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 = [[-1, 0, 1], [3, 2, 13], 18]
After
derived = [[[-1, 3], [0, 2], [1, 13]], 18, 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.
Rule: Legacy_probability_ORIGINAL_rawS_weight_alignment_and_model_qualification · Uses: G001, S003
Before
Legacy probability ORIGINAL included outcome-1 running sum0 := sum(0, 3)
After
Legacy probability ORIGINAL included outcome-1 running sum0 = 3
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, S004
Before
Legacy probability ORIGINAL included outcome0 running sum1 := sum(3, 2)
After
Legacy probability ORIGINAL included outcome0 running sum1 = 5
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, S005
Before
Legacy probability ORIGINAL included outcome1 running sum2 := sum(5, 13)
After
Legacy probability ORIGINAL included outcome1 running sum2 = 18
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, S006
Before
Legacy complement ORIGINAL denominator minus event mass := sub(18, 18)
After
Legacy complement ORIGINAL denominator minus event mass = 0
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, S007
Before
Legacy probability ORIGINAL normalized event or complement fraction := div(0, 18)
After
Legacy probability ORIGINAL normalized event or complement fraction = 0
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, S008
Before
source roles = [[-1, 0, 1], [-1, 0, 1], [[-1, 3], [0, 2], [1, 13]], 18]
After
derived = [18, [], 0, 0/1]
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.
Rule: Legacy_complete_finite_probability_axioms_and_same_carrier_complement · Uses: G001, S009
S010 · premise_and_result_audit — Reverse-check complete original data,operands and premises;finite checks,conditional reasoning,formal proof and publication authority remain distinct.S009 · premise_and_result_audit — Recompute Legacy probability ORIGINAL normalized event or complement fraction with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute Legacy complement ORIGINAL denominator minus event mass with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome1 running sum2 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome0 running sum1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome-1 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.0/1
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: 0b522c285c988b52ce4b902ec3d3bd055855a4c6a871af6beefd5cfa9e7550d9candidate record SHA-256: cb0c9e631698a08be943c4039d56baf4ff75ad19693c8a40c061406dacf3b110semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4eLean: NOT_RUN · human review: NOT_PERFORMED