Question 3144: Comparing Two Directions of Conditional Probability

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0005-03144 · q-20261003-advance-20261004-r07-conditional-probability-0094

Original question and givens

The nonempty finite sample space is S={6, 7, 8, 9}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [0/(27), 3/(27), 5/(27), 19/(27)], summing exactly to one. A={6, 7, 8, 9}, B={7, 8}, 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={6, 7, 8, 9}, with event family F=𝒫(S). In the displayed order of S, the nonnegative normalized weights are [0/(27), 3/(27), 5/(27), 19/(27)], summing exactly to one. A={6, 7, 8, 9}, B={7, 8}, 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

Step-by-step derivation

S001 · implies

Before

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

Rule: conditional_model_domain · Uses: G001

S002 · implies

Before

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

After

A∩B={7, 8}

Conditions: none

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

Step identifiers and dependencies

Rule: explicit_event_intersection · Uses: G001, S001

S003 · calculation

Before

total weight numerator := sum(0, 3, 5, 19)

After

total weight numerator = 27

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

[[6, 0], [7, 3], [8, 5], [9, 19]]

After

weights sum to 27;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 dependencies

Rule: weight_alignment_and_normalization · Uses: G001, S003

S005 · calculation

Before

B weight numerator := sum(3, 5)

After

B weight numerator = 8

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

B probability mass := div(8, 27)

After

B probability mass = 8/27

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

A∩B weight numerator := sum(3, 5)

After

A∩B weight numerator = 8

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

A∩B probability mass := div(8, 27)

After

A∩B probability mass = 8/27

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

P(A|B) := div(8/27, 8/27)

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

P(A∩B)=8/27;P(B)=8/27>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 dependencies

Rule: conditional_ratio_positive_guard · Uses: G001, S009

S011 · calculation

Before

A weight numerator := sum(0, 3, 5, 19)

After

A weight numerator = 27

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

A probability mass := div(27, 27)

After

A probability mass = 1

Conditions: none

Divide by the common positive denominator to obtain exact probability.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

P(B|A) := div(8/27, 1)

After

P(B|A) = 8/27

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

P(A|B)=1;P(B|A)=8/27

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 dependencies

Rule: 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: ef22829ca048e4f6c00f50d2d603dde44f32e7ed557e09cd6f4e5200fdb7b69a
candidate record SHA-256: 83edb823194a3e33c654e1d0af1bbdf147eeb354485425b833200d612ba2383a
semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
Lean: NOT_RUN · human review: NOT_PERFORMED

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