# Question 3129: Comparing Two Directions of Conditional Probability

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0005-03129 · q-20261003-advance-20261004-r07-conditional-probability-0079



## Original question and givens

The nonempty finite sample space is S={-9, -8, -7, -6}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [0/(13), 4/(13), 5/(13), 4/(13)], summing exactly to one. A={-9, -8, -7, -6}, B={-8, -7}, and both P(A) and P(B) are positive. Are P(A|B) and P(B|A) equal?


- G001 — The nonempty finite sample space is S={-9, -8, -7, -6}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [0/(13), 4/(13), 5/(13), 4/(13)], summing exactly to one. A={-9, -8, -7, -6}, B={-8, -7}, and both P(A) and P(B) are positive. Are P(A|B) and P(B|A) equal?





## Definition and scope

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.


- [mko-conditional-probability](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/)





## Step-by-step derivation

S001 · impliesBefore

S={-9, -8, -7, -6};A={-9, -8, -7, -6};B={-8, -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.

Step identifiers and dependenciesRule: conditional_model_domain · Uses: G001





S002 · impliesBefore

A={-9, -8, -7, -6};B={-8, -7}

After

A∩B={-8, -7}

Conditions: none

The intersection keeps outcomes in both events;neither a union nor independence is assumed.

Step identifiers and dependenciesRule: explicit_event_intersection · Uses: G001, S001





S003 · calculationBefore

total weight numerator := sum(0, 4, 5, 4)

After

total weight numerator = 13

Conditions: none

Nonnegative weights must sum to the common positive denominator to give total probability1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

[[-9, 0], [-8, 4], [-7, 5], [-6, 4]]

After

weights sum to 13;normalization verified

Conditions: none

Weights align with the original S order;do not sort S while leaving weights in their old positions.

Step identifiers and dependenciesRule: weight_alignment_and_normalization · Uses: G001, S003





S005 · calculationBefore

B weight numerator := sum(4, 5)

After

B weight numerator = 9

Conditions: none

Sum aligned weights over the event;zero-mass outcomes do not become positive mass.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

B probability mass := div(9, 13)

After

B probability mass = 9/13

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

A∩B weight numerator := sum(4, 5)

After

A∩B weight numerator = 9

Conditions: none

Sum aligned weights over the event;zero-mass outcomes do not become positive mass.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

A∩B probability mass := div(9, 13)

After

A∩B probability mass = 9/13

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

P(A|B) := div(9/13, 9/13)

After

P(A|B) = 1

Conditions: none

By definition divide the intersection probability by positive P(B);do not assume it equals P(A)P(B).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · impliesBefore

P(A∩B)=9/13;P(B)=9/13>0

After

P(A|B)=1

Conditions: none

Use the ratio only with the checked positive conditioning mass;a zero-mass intersection can validly yield conditional value0.

Step identifiers and dependenciesRule: conditional_ratio_positive_guard · Uses: G001, S009





S011 · calculationBefore

A weight numerator := sum(0, 4, 5, 4)

After

A weight numerator = 13

Conditions: none

Sum aligned weights over the event;zero-mass outcomes do not become positive mass.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

A probability mass := div(13, 13)

After

A probability mass = 1

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

P(B|A) := div(9/13, 1)

After

P(B|A) = 9/13

Conditions: none

The reverse direction uses positive P(A),not P(B).

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · impliesBefore

P(A|B)=1;P(B|A)=9/13

After

equal=No

Conditions: none

Equality of the two directions is only a numerical relation,not independence.Independence instead requires P(A∩B)=P(A)P(B);disjoint positive-probability events have both conditionals0 but are dependent.

Step identifiers and dependenciesRule: conditional_direction_not_independence · Uses: G001, S013









## Reverse checks


- S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S013 · premise_and_result_audit — Recompute P(B|A) with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute A probability mass with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute A weight numerator with exact rational arithmetic, not rounded decimals.
- 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.





## Result

No

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: bddfd731b6872eed239761230e03411b8bf8f949b788b279b3d1b5410bf2666e
- candidate record SHA-256: 4f90588bb7bba1265cbd4a58f3cc7983b610020e03ffe3fc1e0905f0b79dc8c0
- semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
- question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
- Lean: NOT_RUN · human review: NOT_PERFORMED

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