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R0005-03162 · q-20261003-advance-20261004-r07-conditional-probability-0062
The nonempty finite sample space is S={-1, 0, 1, 2}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(21), 4/(21), 3/(21), 12/(21)], summing exactly to one. Does B={-1, 0, 1, 2} meet the positive-probability requirement for defining P(A|B)?
G001 — The nonempty finite sample space is S={-1, 0, 1, 2}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(21), 4/(21), 3/(21), 12/(21)], summing exactly to one. Does B={-1, 0, 1, 2} meet the positive-probability requirement for defining P(A|B)?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
S={-1, 0, 1, 2};no particular A supplied;B={-1, 0, 1, 2};F=𝒫(S)After
events are declared subsets;conditioning needs P(B)>0
Conditions: none
𝒫(S) is the full event powerset,not a probability value.A nonempty event may have probability0.When checking B's eligibility,do not invent an unspecified A.
Rule: conditional_model_domain · Uses: G001
Before
total weight numerator := sum(2, 4, 3, 12)
After
total weight numerator = 21
Conditions: none
Nonnegative weights must sum to the common positive denominator to give total probability1.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
[[-1, 2], [0, 4], [1, 3], [2, 12]]
After
weights sum to 21;normalization verified
Conditions: none
Weights align with the original S order;do not sort S while leaving weights in their old positions.
Rule: weight_alignment_and_normalization · Uses: G001, S002
Before
B weight numerator := sum(2, 4, 3, 12)
After
B weight numerator = 21
Conditions: none
Sum aligned weights over the event;zero-mass outcomes do not become positive mass.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
B probability mass := div(21, 21)
After
B probability mass = 1
Conditions: none
Divide by the common positive denominator to obtain exact probability.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
P(B)=1
After
conditioning eligible=Yes
Conditions: none
The ordinary conditional ratio requires P(B)>0;when P(B)=0 do not assign0 to an undefined ratio.
Rule: positive_conditioning_eligibility · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute B probability mass with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute B weight numerator with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Recompute total weight numerator with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.Yes
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: 290ed11bc3dd9c08667ab32d2374f641c6bdb45a6d1cf10f32d933ce6ae2d334candidate record SHA-256: d57ed80b0b137ca03283ccdfd76f2f00d7be4bbd2b50d8cb132f3a34cc9d2ec0semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9Lean: NOT_RUN · human review: NOT_PERFORMED