Question 4166: Positive-Mass Value Recovered from a Mean

Original question and givens

S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 17, -13] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-219/19. Find that slot’s integer value, written as p/1.

• G001 — S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 17, -13] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-219/19. Find that slot’s integer value, written as p/1.

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-expectation

Step-by-step derivation

total mass numerator := sum(16, 2, 1)

total mass numerator = 19

indexed masses=[16, 2, 1];D=19

probabilities=[16/19, 2/19, 1/19]

known numerator1 := mul(2, 17)

known numerator1 = 34

known numerator2 := mul(1, -13)

known numerator2 = -13

known weighted numerator := sum(34, -13)

known weighted numerator = 21

target expectation numerator := mul(-219/19, 19)

target expectation numerator = -219

unknown weighted contribution := sub(-219, 21)

unknown weighted contribution = -240

recovered unknown value := div(-240, 16)

recovered unknown value = -15

recovered weighted numerator := mul(16, -15)

recovered weighted numerator = -240

complete reconstructed numerator := sum(21, -240)

complete reconstructed numerator = -219

reconstructed expectation := div(-219, 19)

reconstructed expectation = -219/19

k=0;mass_k=16>0;target=-219/19

unique recovered value=-15;substitution mean=-219/19

Reverse checks

• 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 — Recompute reconstructed expectation with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute complete reconstructed numerator with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute recovered weighted numerator with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute recovered unknown value with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute unknown weighted contribution with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute target expectation numerator with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute known weighted numerator with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute known numerator2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute known numerator1 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 — Recompute total mass numerator with exact rational arithmetic, not rounded decimals.

Result

-15/1

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.