Question 4559: First Mismatch on Positive Probability Support

Original question and givens

S={0,…,7}, F=P(S), masses=[9, 0, 2, 0, 0, 0, 0, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 9, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

• G001 — S={0,…,7}, F=P(S), masses=[9, 0, 2, 0, 0, 0, 0, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 9, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

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-variance

Step-by-step derivation

total mass numerator := sum(9, 0, 2, 0, 0, 0, 0, 0)

total mass numerator = 11

probability weight0 := div(9, 11)

probability weight0 = 9/11

probability weight1 := div(0, 11)

probability weight1 = 0

probability weight2 := div(2, 11)

probability weight2 = 2/11

probability weight3 := div(0, 11)

probability weight3 = 0

probability weight4 := div(0, 11)

probability weight4 = 0

probability weight5 := div(0, 11)

probability weight5 = 0

probability weight6 := div(0, 11)

probability weight6 = 0

probability weight7 := div(0, 11)

probability weight7 = 0

values=[0, 101, 9, 103, 104, 105, 106, 107];weights=[9/11, 0/1, 2/11, 0/1, 0/1, 0/1, 0/1, 0/1]

known finite probability law,not a sampled estimate

index=0;mass=9;X=0;anchor X=0

positive-mass mismatch=No

index=1;mass=0;X=101;anchor X=0

positive-mass mismatch=No

index=2;mass=2;X=9;anchor X=0

positive-mass mismatch=Yes

index=3;mass=0;X=103;anchor X=0

positive-mass mismatch=No

index=4;mass=0;X=104;anchor X=0

positive-mass mismatch=No

index=5;mass=0;X=105;anchor X=0

positive-mass mismatch=No

index=6;mass=0;X=106;anchor X=0

positive-mass mismatch=No

index=7;mass=0;X=107;anchor X=0

positive-mass mismatch=No

first positive index=0;all eligible mismatch indices=[2]

2

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 probability weight7 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute probability weight6 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute probability weight5 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute probability weight4 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute probability weight3 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute probability weight2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute probability weight1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute probability weight0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals.

Result

2

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.