# Question 30120: Complete Pointwise Conditional Law [Return to catalogue page 151](https://ocme.evemisslab.com/en/questions/page/151/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r04-conditional-probability-foundations/#q-20261008-advance-20261010-r04-conditional-probability-0070) 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. R0059-30120 · q-20261008-advance-20261010-r04-conditional-probability-0070 ## Original question and givens S={0,…,3}, F=P(S), masses=[19, 1, 2, 3], D=25, 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=[19, 1, 2, 3], D=25, 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](https://ocme.evemisslab.com/en/mko/mko-conditional-probability/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Conditional ORIGINAL ALL numerator weights sum := sum(19, 1, 2, 3) After Conditional ORIGINAL ALL numerator weights sum = 25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [[19, 1, 2, 3], 25] After derived = [4, [[0, 19/25], [1, 1/25], [2, 2/25], [3, 3/25]], 25] 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 dependenciesRule: Conditional_FULL_normalized_finite_law_not_candidate_repair · Uses: G001, S002 S004 · impliesBefore 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 dependenciesRule: Conditional_RAW_event_bound_BEFORE_dedup_and_Fin_index_typing · Uses: G001, S003 S005 · calculationBefore Conditional B ORIGINAL included index0 running0 := sum(0, 19) After Conditional B ORIGINAL included index0 running0 = 19 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Conditional B ORIGINAL included index1 running1 := sum(19, 1) After Conditional B ORIGINAL included index1 running1 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Conditional B ORIGINAL included index2 running2 := sum(20, 2) After Conditional B ORIGINAL included index2 running2 = 22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [Conditional B, [0, 1, 2], [19, 1, 2, 3]] After derived = [22] 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 dependenciesRule: Conditional_COMPLETE_subset_mass_in_ORIGINAL_slots · Uses: G001, S007 S009 · calculationBefore Conditional ORIGINAL B mass over D := div(22, 25) After Conditional ORIGINAL B mass over D = 22/25 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [[0, 1, 2], [19, 1, 2, 3], 25] After derived = [22, 22/25, 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 dependenciesRule: Conditional_STRICT_positive_B_mass_not_merely_nonempty · Uses: G001, S009 S011 · calculationBefore Conditional ORIGINAL singleton0 over B mass := div(19, 22) After Conditional ORIGINAL singleton0 over B mass = 19/22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Conditional ORIGINAL singleton1 over B mass := div(1, 22) After Conditional ORIGINAL singleton1 over B mass = 1/22 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Conditional ORIGINAL singleton2 over B mass := div(2, 22) After Conditional ORIGINAL singleton2 over B mass = 1/11 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Conditional ORIGINAL singleton3 over B mass := div(0, 22) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Conditional reverse ALL singleton probabilities sum := sum(19/22, 1/22, 1/11, 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore source roles = [[19, 1, 2, 3], 25, [0, 1, 2], 22] After derived = [[19/22, 1/22, 1/11, 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 dependenciesRule: 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 [19/22, 1/22, 1/11, 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: 0340783f49a2e51d9f34bebfd64f084975dd0fd3e4122f41e7f20a591b3618a8 - candidate record SHA-256: 3a15fcc509e360db136e3ca7e5ffb074a220e4256dff161a0c1c21b515905014 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r04-conditional-probability-0070.json)