Explore related knowledge and questions →
← Question derivation index · Original question batch
R0055-28302 · q-20261003-r07-probability-0077
The nonempty finite sample space is S={-11, -10, -9} and the event family is F=𝒫(S). In the displayed order of S, the weights are [1/(6), 2/(6), 3/(6)] and E={-11, -10, -9}. Find P(S∖E) exactly as a reduced fraction with positive denominator; do not assume equiprobability.
G001 — The nonempty finite sample space is S={-11, -10, -9} and the event family is F=𝒫(S). In the displayed order of S, the weights are [1/(6), 2/(6), 3/(6)] and E={-11, -10, -9}. 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 = [[-11, -10, -9], powerset(S)]
After
derived = [[-11, -10, -9], 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(1, 2, 3)
After
Legacy probability ORIGINAL ALL numerator weights sum = 6
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 = [[-11, -10, -9], [1, 2, 3], 6]
After
derived = [[[-11, 1], [-10, 2], [-9, 3]], 6, 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-11 running sum0 := sum(0, 1)
After
Legacy probability ORIGINAL included outcome-11 running sum0 = 1
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 outcome-10 running sum1 := sum(1, 2)
After
Legacy probability ORIGINAL included outcome-10 running sum1 = 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, S005
Before
Legacy probability ORIGINAL included outcome-9 running sum2 := sum(3, 3)
After
Legacy probability ORIGINAL included outcome-9 running sum2 = 6
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(6, 6)
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, 6)
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 = [[-11, -10, -9], [-11, -10, -9], [[-11, 1], [-10, 2], [-9, 3]], 6]
After
derived = [6, [], 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 outcome-9 running sum2 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome-10 running sum1 with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute Legacy probability ORIGINAL included outcome-11 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: c34b17d027b598fd66d3b4146d2430cea26c1a840ab2c067a9d94f25a2ac2319candidate record SHA-256: 36ba8f416559314026e787d655df3e6cd7448ea255b4538be601c202901c8c89semantic review receipt SHA-256: 6af367dae24a584f3596525c4953bbc2115343efd68cc9083693eb5c1ef2e683question-first witness SHA-256: 611eb69f3716404b968296a8b0d7d8c61b5ca18129d406f1ff9c7c11e10ffb4eLean: NOT_RUN · human review: NOT_PERFORMED