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Question 30049: Conditional Candidate Eligibility and Signed Defect

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0059-30049 · q-20261008-advance-20261010-r04-conditional-probability-0099

Original question and givens

S={0,…,3}, F=P(S), masses=[23, 1, 2, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=22/24. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.

  • G001 — S={0,…,3}, F=P(S), masses=[23, 1, 2, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. A=[0, 2], B=[0, 1], P(B)>0. Candidate=22/24. Give (does it equal P(A|B)?, candidate−P(A|B)). A candidate outside [0,1] may be submitted for rejection.

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-density10-conditional-candidate-defect, density10_conditional_candidate_defect, finite_normalized_probability]

After

derived = [[candidate_matches, SIGNED_candidate_minus_actual]]

Conditions: none

Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.

Step identifiers and dependencies

Rule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001

S002 · calculation

Before

Conditional ORIGINAL ALL numerator weights sum := sum(23, 1, 2, 0)

After

Conditional ORIGINAL ALL numerator weights sum = 26

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [[23, 1, 2, 0], 26]

After

derived = [4, [[0, 23/26], [1, 1/26], [2, 1/13], [3, 0/1]], 26]

Conditions: none

On S=Fin(n) with its full powerset,each singleton has mass m_i/D,all m_i>=0 and total D>0.These FOUR families REQUIRE an actual normalized law;reject an invalid law rather than return No and repair its weights.

Step identifiers and dependencies

Rule: Conditional_FULL_normalized_finite_law_not_candidate_repair · Uses: G001, S002

S004 · implies

Before

source roles = [B, [0, 1], 4]

After

derived = [[0, 1]]

Conditions: none

FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.

Step identifiers and dependencies

Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S003

S005 · calculation

Before

Conditional B ORIGINAL included index0 running0 := sum(0, 23)

After

Conditional B ORIGINAL included index0 running0 = 23

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Conditional B ORIGINAL included index1 running1 := sum(23, 1)

After

Conditional B ORIGINAL included index1 running1 = 24

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Conditional B, [0, 1], [23, 1, 2, 0]]

After

derived = [24]

Conditions: none

For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.

Step identifiers and dependencies

Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S006

S008 · calculation

Before

Conditional ORIGINAL B mass over D := div(24, 26)

After

Conditional ORIGINAL B mass over D = 12/13

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [[0, 1], [23, 1, 2, 0], 26]

After

derived = [24, 12/13, Yes]

Conditions: none

P(B)>0 means b>0,not merely B nonempty.A nonempty event containing only null labels cannot be conditioned on.Divide by b ONLY under the given positive-mass premise.

Step identifiers and dependencies

Rule: Conditional_STRICT_positive_B_mass_not_merely_nonempty · Uses: G001, S008

S010 · implies

Before

source roles = [A, [0, 2], 4]

After

derived = [[0, 2]]

Conditions: none

FIRST require raw length<=16 and each SAFE index in Fin(n),THEN deduplicate and sort numerically.Repeated listings do not multiply mass;17 raw copies still exceed the source bound.

Step identifiers and dependencies

Rule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S009

S011 · calculation

Before

Conditional intersection ORIGINAL included index0 running0 := sum(0, 23)

After

Conditional intersection ORIGINAL included index0 running0 = 23

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [Conditional intersection, [0], [23, 1, 2, 0]]

After

derived = [23]

Conditions: none

For each normalized event index take its ORIGINAL slot exactly once.An empty event has sum0;do not invent uniform weights or change alignment.

Step identifiers and dependencies

Rule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S011

S013 · calculation

Before

Conditional ORIGINAL intersection mass over D := div(23, 26)

After

Conditional ORIGINAL intersection mass over D = 23/26

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

Conditional ORIGINAL intersection over B mass := div(23, 24)

After

Conditional ORIGINAL intersection over B mass = 23/24

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

Conditional reverse probability ratio SAME law := div(23/26, 12/13)

After

Conditional reverse probability ratio SAME law = 23/24

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [23, 24, 26, 23/26, 12/13]

After

derived = [23/24, 23/24]

Conditions: none

P(A|B)=P(A intersection B)/P(B)=(i/D)/(b/D)=i/b,cancelling D from the SAME law.No independence is assumed,and an instance equality does not imply general independence.

Step identifiers and dependencies

Rule: Conditional_RATIO_cancel_SAME_positive_D_and_B · Uses: G001, S015

S017 · calculation

Before

Conditional ORIGINAL SIGNED candidate minus actual := sub(11/12, 23/24)

After

Conditional ORIGINAL SIGNED candidate minus actual = -1/24

Conditions: none

Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [22, 24, 11/12, 23/24]

After

derived = [No, -1/24]

Conditions: none

A negative or greater-than1 candidate is a LEGAL audit input.Match is Yes ONLY when candidate=actual.Return SIGNED candidate-actual,not its absolute value;an incorrect candidate does not invalidate the underlying law.

Step identifiers and dependencies

Rule: Conditional_candidate_OUTSIDE_probability_range_LEGAL_signed_defect · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S017 · premise_and_result_audit — Recompute Conditional ORIGINAL SIGNED candidate minus actual with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S015 · premise_and_result_audit — Recompute Conditional reverse probability ratio SAME law with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Conditional ORIGINAL intersection over B mass with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Conditional ORIGINAL intersection mass over D with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S011 · premise_and_result_audit — Recompute Conditional intersection ORIGINAL included index0 running0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S009 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S008 · premise_and_result_audit — Recompute Conditional ORIGINAL B mass over D with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S006 · premise_and_result_audit — Recompute Conditional B ORIGINAL included index1 running1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Conditional B ORIGINAL included index0 running0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S003 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
  • S002 · premise_and_result_audit — Recompute Conditional ORIGINAL ALL numerator weights sum with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.

Result

(No, -1/24)

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: 13f4d8ef134340fd1d940b3805d436bf52fae21b59707662e55c86055ec2e37b
  • candidate record SHA-256: e462be69ffbfc732994a579bd97e689ef31e96999894bda53b1c3d0386e9b328
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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