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R0056-28545 · q-20261004-advance-20261005-r05-random-variable-0095
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, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) 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, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.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.
Before
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.
Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001
Before
source roles = [[0, 1, 2, 3], powerset(S), [-20, -20, 21, 1018]]
After
derived = [[[0, -20], [1, -20], [2, 21], [3, 1018]]]
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.
Rule: RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment · Uses: G001, S001
Before
RV ORIGINAL all aligned numerator weights := sum(1, 21, 3, 0)
After
RV ORIGINAL all aligned numerator weights = 25
Conditions: none
Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
source roles = [[[0, -20, 1], [1, -20, 21], [2, 21, 3], [3, 1018, 0]], 25, 0]
After
derived = [25, [[0, -20, 1, No], [1, -20, 21, No], [2, 21, 3, No], [3, 1018, 0, No]]]
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.
Rule: RV_point_mass_normalized_law_ON_S_and_ALL_fibre_terms · Uses: G001, S003
Before
RV ORIGINAL complete fibre mass over denominator := div(0, 25)
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.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
source roles = [[[0, -20, 1], [1, -20, 21], [2, 21, 3], [3, 1018, 0]], 0, 0, 25]
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.
Rule: RV_point_mass_unattained_value_zero_and_reduced_fraction · Uses: G001, S005
S006 · 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.S005 · premise_and_result_audit — Recompute RV ORIGINAL complete fibre mass over denominator 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.0/1
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 869b4dba3f1fe262e3ab2a2265c936602040e5bf383dd54b13c8d2bb0cb5faa5candidate record SHA-256: 691ee0c44b0355b1231b9d16d1cd8127b943e0c7bfb9d28451fd64100d1030e5semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759fLean: NOT_RUN · human review: NOT_PERFORMED