# Question 28532: Finite Point Probability of a Repeated Value [Return to catalogue page 143](https://ocme.evemisslab.com/en/questions/page/143/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r05-random-variable-foundations/#q-20261004-advance-20261005-r05-random-variable-0082) 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. R0056-28532 · q-20261004-advance-20261005-r05-random-variable-0082 ## Original question and givens The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 8), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 8/(12), 3/(12), 0/(12)], summing exactly to one. Find P(X=1005) exactly as a reduced fraction p/q with positive denominator. - G001 — The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 8), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 8/(12), 3/(12), 0/(12)], summing exactly to one. Find P(X=1005) exactly as a reduced fraction p/q with positive denominator. ## 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-random-variable](https://ocme.evemisslab.com/en/mko/mko-random-variable/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-random-variable-point-mass, rv_point_mass] After derived = [probability_of_complete_value_fibre] Conditions: none Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions. Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001 S002 · impliesBefore source roles = [[0, 1, 2, 3], powerset(S), [-7, -7, 8, 1005]] After derived = [[[0, -7], [1, -7], [2, 8], [3, 1005]]] Conditions: none Align each value with the ORIGINAL S slot,not independently sorted columns.The full powerset contains every preimage of this real-valued function,hence measurability.No invented uniform weights are needed for a value,range or event. Step identifiers and dependenciesRule: RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment · Uses: G001, S001 S003 · calculationBefore RV ORIGINAL all aligned numerator weights := sum(1, 8, 3, 0) After RV ORIGINAL all aligned numerator weights = 12 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [[[0, -7, 1], [1, -7, 8], [2, 8, 3], [3, 1005, 0]], 12, 1005] After derived = [12, [[0, -7, 1, No], [1, -7, 8, No], [2, 8, 3, No], [3, 1005, 0, Yes]]] Conditions: none Nonnegative weights on the SAME S must sum to den>0.P(X=t) adds the COMPLETE fibre,including repeated X values and zero weights;range size is not the denominator. Step identifiers and dependenciesRule: RV_point_mass_normalized_law_ON_S_and_ALL_fibre_terms · Uses: G001, S003 S005 · calculationBefore RV ORIGINAL matching slot3 outcome3 mass := sum(0, 0) After RV ORIGINAL matching slot3 outcome3 mass = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore RV ORIGINAL complete fibre mass over denominator := div(0, 12) After RV ORIGINAL complete fibre mass over denominator = 0 Conditions: none Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [[[0, -7, 1], [1, -7, 8], [2, 8, 3], [3, 1005, 0]], 1005, 0, 12] After derived = [0/1] Conditions: none An unattained value has empty-fibre sum0 and probability0/1.Otherwise add all matching terms and divide by the ORIGINAL denominator.Reduction preserves the event;integer probabilities retain/1. Step identifiers and dependenciesRule: RV_point_mass_unattained_value_zero_and_reduced_fraction · Uses: G001, S006 ## Reverse checks - S007 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S006 · premise_and_result_audit — Recompute RV ORIGINAL complete fibre mass over denominator with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute RV ORIGINAL matching slot3 outcome3 mass with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S003 · premise_and_result_audit — Recompute RV ORIGINAL all aligned numerator weights with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. - S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance. ## Result 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: 8a0e1acd8169b72505f6880355f4116c0129eb166c7d709b7f68c0ab089f6680 - candidate record SHA-256: fbee804c51b71cd781cad9b5c99acd9fc1d264af31cfa5feee6ba17ce51c4e3f - semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8 - question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r05-random-variable-0082.json)