Question 30115: Complete Pointwise Conditional Law

Original question and givens

S={0,…,3}, F=P(S), masses=[14, 1, 2, 3], D=20, 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=[14, 1, 2, 3], D=20, 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.

• mko-conditional-probability

Step-by-step derivation

source roles = [qf-density10-conditional-point-law, density10_conditional_point_law, finite_normalized_probability]

derived = [[ALL_singleton_conditional_probabilities_index_order]]

Conditional ORIGINAL ALL numerator weights sum := sum(14, 1, 2, 3)

Conditional ORIGINAL ALL numerator weights sum = 20

source roles = [[14, 1, 2, 3], 20]

derived = [4, [[0, 7/10], [1, 1/20], [2, 1/10], [3, 3/20]], 20]

source roles = [B, [0, 1, 2], 4]

derived = [[0, 1, 2]]

Conditional B ORIGINAL included index0 running0 := sum(0, 14)

Conditional B ORIGINAL included index0 running0 = 14

Conditional B ORIGINAL included index1 running1 := sum(14, 1)

Conditional B ORIGINAL included index1 running1 = 15

Conditional B ORIGINAL included index2 running2 := sum(15, 2)

Conditional B ORIGINAL included index2 running2 = 17

source roles = [Conditional B, [0, 1, 2], [14, 1, 2, 3]]

derived = [17]

Conditional ORIGINAL B mass over D := div(17, 20)

Conditional ORIGINAL B mass over D = 17/20

source roles = [[0, 1, 2], [14, 1, 2, 3], 20]

derived = [17, 17/20, Yes]

Conditional ORIGINAL singleton0 over B mass := div(14, 17)

Conditional ORIGINAL singleton0 over B mass = 14/17

Conditional ORIGINAL singleton1 over B mass := div(1, 17)

Conditional ORIGINAL singleton1 over B mass = 1/17

Conditional ORIGINAL singleton2 over B mass := div(2, 17)

Conditional ORIGINAL singleton2 over B mass = 2/17

Conditional ORIGINAL singleton3 over B mass := div(0, 17)

Conditional ORIGINAL singleton3 over B mass = 0

Conditional reverse ALL singleton probabilities sum := sum(14/17, 1/17, 2/17, 0)

Conditional reverse ALL singleton probabilities sum = 1

source roles = [[14, 1, 2, 3], 20, [0, 1, 2], 17]

derived = [[14/17, 1/17, 2/17, 0/1], 1/1]

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

[14/17, 1/17, 2/17, 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.