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Question 30121: Complete Pointwise Conditional Law

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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R0059-30121 · q-20261008-advance-20261010-r04-conditional-probability-0071

Original question and givens

S={0,…,3}, F=P(S), masses=[20, 1, 2, 3], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.

  • G001 — S={0,…,3}, F=P(S), masses=[20, 1, 2, 3], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. B=[0, 1, 2], P(B)>0. In index order from 0, list the conditional probability of every singleton given B, including zero terms outside B. Use exact reduced fractions with positive denominators, retaining /1 for integers.

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-point-law, density10_conditional_point_law, finite_normalized_probability]

After

derived = [[ALL_singleton_conditional_probabilities_index_order]]

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(20, 1, 2, 3)

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 = [[20, 1, 2, 3], 26]

After

derived = [4, [[0, 10/13], [1, 1/26], [2, 1/13], [3, 3/26]], 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, 2], 4]

After

derived = [[0, 1, 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, S003

S005 · calculation

Before

Conditional B ORIGINAL included index0 running0 := sum(0, 20)

After

Conditional B ORIGINAL included index0 running0 = 20

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(20, 1)

After

Conditional B ORIGINAL included index1 running1 = 21

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 · calculation

Before

Conditional B ORIGINAL included index2 running2 := sum(21, 2)

After

Conditional B ORIGINAL included index2 running2 = 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, S006

S008 · implies

Before

source roles = [Conditional B, [0, 1, 2], [20, 1, 2, 3]]

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, S007

S009 · calculation

Before

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

After

Conditional ORIGINAL B 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, S008

S010 · implies

Before

source roles = [[0, 1, 2], [20, 1, 2, 3], 26]

After

derived = [23, 23/26, 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, S009

S011 · calculation

Before

Conditional ORIGINAL singleton0 over B mass := div(20, 23)

After

Conditional ORIGINAL singleton0 over B mass = 20/23

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 · calculation

Before

Conditional ORIGINAL singleton1 over B mass := div(1, 23)

After

Conditional ORIGINAL singleton1 over B mass = 1/23

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, S011

S013 · calculation

Before

Conditional ORIGINAL singleton2 over B mass := div(2, 23)

After

Conditional ORIGINAL singleton2 over B mass = 2/23

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, S012

S014 · calculation

Before

Conditional ORIGINAL singleton3 over B mass := div(0, 23)

After

Conditional ORIGINAL singleton3 over B mass = 0

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 ALL singleton probabilities sum := sum(20/23, 1/23, 2/23, 0)

After

Conditional reverse ALL singleton probabilities sum = 1

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 = [[20, 1, 2, 3], 26, [0, 1, 2], 23]

After

derived = [[20/23, 1/23, 2/23, 0/1], 1/1]

Conditions: none

For EVERY original index list P({i}|B)=m_i/b or0/1,including outside-B and inside-B null labels.The COMPLETE vector sums to b/b=1.Do not drop zero entries or change indexing.

Step identifiers and dependencies

Rule: Conditional_POINT_law_ALL_indices_NULL_and_outside_ZERO · Uses: G001, S015

Reverse checks

  • 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 ALL singleton probabilities sum with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute Conditional ORIGINAL singleton3 over B mass with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Recompute Conditional ORIGINAL singleton2 over B mass with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute Conditional ORIGINAL singleton1 over B mass with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Conditional ORIGINAL singleton0 over B mass 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 — Recompute Conditional ORIGINAL B mass over D with exact rational arithmetic, not rounded decimals.
  • S008 · 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.
  • S007 · premise_and_result_audit — Recompute Conditional B ORIGINAL included index2 running2 with exact rational arithmetic, not rounded decimals.
  • 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

[20/23, 1/23, 2/23, 0/1]

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: cee4f65c68c19c7b1e77f6fae5af6131bfe0b99fb0b7593e274aa6541c55c9f1
  • candidate record SHA-256: e176666bbb4397c8d26d4381eda49ba0962abb80d91ea779581609060dc24bca
  • semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5
  • question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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