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R0005-03218 · q-20261003-advance-20261004-r07-conditional-probability-0018
The nonempty finite sample space is S={5, 6, 7, 8}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(26), 2/(26), 4/(26), 18/(26)], summing exactly to one. A={5, 6}, B={6, 7}, and P(B)>0. Compute P(A|B) exactly as a reduced fraction with positive denominator, without assuming equiprobability.
G001 — The nonempty finite sample space is S={5, 6, 7, 8}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [2/(26), 2/(26), 4/(26), 18/(26)], summing exactly to one. A={5, 6}, B={6, 7}, and P(B)>0. Compute P(A|B) exactly as a reduced fraction with positive denominator, without assuming 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
S={5, 6, 7, 8};A={5, 6};B={6, 7};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
A={5, 6};B={6, 7}After
A∩B={6}Conditions: none
The intersection keeps outcomes in both events;neither a union nor independence is assumed.
Rule: explicit_event_intersection · Uses: G001, S001
Before
total weight numerator := sum(2, 2, 4, 18)
After
total weight numerator = 26
Conditions: none
Nonnegative weights must sum to the common positive denominator to give total probability1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
[[5, 2], [6, 2], [7, 4], [8, 18]]
After
weights sum to 26;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, S003
Before
B weight numerator := sum(2, 4)
After
B weight numerator = 6
Conditions: none
Sum aligned weights over the event;zero-mass outcomes do not become positive mass.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
B probability mass := div(6, 26)
After
B probability mass = 3/13
Conditions: none
Divide by the common positive denominator to obtain exact probability.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
A∩B weight numerator := sum(2)
After
A∩B weight numerator = 2
Conditions: none
Sum aligned weights over the event;zero-mass outcomes do not become positive mass.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
A∩B probability mass := div(2, 26)
After
A∩B probability mass = 1/13
Conditions: none
Divide by the common positive denominator to obtain exact probability.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
P(A|B) := div(1/13, 3/13)
After
P(A|B) = 1/3
Conditions: none
By definition divide the intersection probability by positive P(B);do not assume it equals P(A)P(B).
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
P(A∩B)=1/13;P(B)=3/13>0
After
P(A|B)=1/3
Conditions: none
Use the ratio only with the checked positive conditioning mass;a zero-mass intersection can validly yield conditional value0.
Rule: conditional_ratio_positive_guard · Uses: G001, S009
S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute P(A|B) with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Recompute A∩B probability mass with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute A∩B weight numerator with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute B probability mass with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute B weight numerator with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute total weight numerator with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.1/3
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.
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