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R0005-03223 · q-20261003-advance-20261004-r07-conditional-probability-0023
The nonempty finite sample space is S={10, 11, 12, 13}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [1/(31), 3/(31), 4/(31), 23/(31)], summing exactly to one. A={10, 11}, B={11, 12}, 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={10, 11, 12, 13}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [1/(31), 3/(31), 4/(31), 23/(31)], summing exactly to one. A={10, 11}, B={11, 12}, 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={10, 11, 12, 13};A={10, 11};B={11, 12};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={10, 11};B={11, 12}After
A∩B={11}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(1, 3, 4, 23)
After
total weight numerator = 31
Conditions: none
Nonnegative weights must sum to the common positive denominator to give total probability1.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
[[10, 1], [11, 3], [12, 4], [13, 23]]
After
weights sum to 31;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(3, 4)
After
B weight numerator = 7
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(7, 31)
After
B probability mass = 7/31
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(3)
After
A∩B weight numerator = 3
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(3, 31)
After
A∩B probability mass = 3/31
Conditions: none
Divide by the common positive denominator to obtain exact probability.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
P(A|B) := div(3/31, 7/31)
After
P(A|B) = 3/7
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)=3/31;P(B)=7/31>0
After
P(A|B)=3/7
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.3/7
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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