# Question 12923: Test Independence by the Exact Product Definition [Return to catalogue page 65](https://ocme.evemisslab.com/en/questions/page/65/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261027-r06-event-independence-foundations/#q-20261014-advance-20261027-r06-event-independence-0023) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0024-12923 · q-20261014-advance-20261027-r06-event-independence-0023 ## Original question and givens Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?). - G001 — Use one complete finite probability model with aligned exact masses=[8/49, 20/49, 6/49, 15/49] on Fin(4)={0,…,3}. Masses must be nonnegative and sum exactly to 1; do not assume uniformity. Events A=[2, 3], B=[1, 3] are carrier-label sets, normalized by removing duplicates and sorting numerically. The primary definition is P(A∩B)=P(A)P(B); conditional formulas require positive conditioning probability. Only two-event independence is checked, not mutual independence of a larger family. Use exact reduced rational outputs with positive denominators; omit /1 for integers. Give (normalized A,normalized B,P_A,P_B,P_intersection,product,independent?). ## 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-event-independence](https://ocme.evemisslab.com/en/mko/mko-event-independence/) ## Step-by-step derivation S001 · calculationBefore event model whole mass total := sum(8/49, 20/49, 6/49, 15/49) After event model whole mass total = 1 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore complete carrier Fin(4);raw aligned masses=[[8, 49], [20, 49], [6, 49], [15, 49]] After normalized masses=[8/49, 20/49, 6/49, 15/49];nonnegative=Yes;total=1 Conditions: none Qualify one complete finite PMF with nonnegative atoms and exact total1.Zero-mass atoms remain in the carrier;complements are not taken only within positive-mass support. Step identifiers and dependenciesRule: event_common_complete_probability_space · Uses: G001, S001 S003 · impliesBefore raw carrier events A=[2, 3];B=[1, 3];carrier size=4 After normalized A=[2, 3];B=[1, 3];intersection=[3] Conditions: none Events are carrier-index sets,normalized without dropping zero-mass labels.Intersection requires the same label in A and B.This differs from coordinate-value events in a joint table. Step identifiers and dependenciesRule: event_carrier_label_sets_and_intersection · Uses: G001, S002 S004 · calculationBefore A mass := sum(6/49, 15/49) After A mass = 3/7 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore B mass := sum(20/49, 15/49) After B mass = 5/7 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore A intersection B mass := sum(15/49) After A intersection B mass = 15/49 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore event product P_A times P_B := mul(3/7, 5/7) After event product P_A times P_B = 15/49 Conditions: none Compute exactly from the original roles and complete established operands,without changing pairings,order or replacing guards by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore P_A=3/7;P_B=5/7;P_intersection=15/49;product=15/49 After two-event independent=Yes Conditions: none The primary definition is intersection probability equal to the marginal product,not disjointness.Do not assume independence before multiplying.This checks two events,not mutual independence of a larger family. Step identifiers and dependenciesRule: independence_primary_product_definition · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute event product P_A times P_B with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute A intersection B mass with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute B mass with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute A mass with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 event model whole mass total with exact rational arithmetic, not rounded decimals. ## Result ([2, 3], [1, 3], 3/7, 5/7, 15/49, 15/49, Yes) 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: 1297d8485313c2571f2c4801ed07d1368e337431a3bd775ee37cc4fe55ddd8f0 - candidate record SHA-256: e77c850ac117460295337743bfab7275fb70e3a4775082d5c43bfe69f16910a4 - semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35 - question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r06-event-independence-0023.json)