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Question 28546: Finite Point Probability of a Repeated Value

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0056-28546 · q-20261004-advance-20261005-r05-random-variable-0096

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, -21), (1, -21), (2, 22), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Find P(X=-21) 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, -21), (1, -21), (2, 22), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Find P(X=-21) 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.

Step-by-step derivation

S001 · implies

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.

Step identifiers and dependencies

Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001

S002 · implies

Before

source roles = [[0, 1, 2, 3], powerset(S), [-21, -21, 22, 1019]]

After

derived = [[[0, -21], [1, -21], [2, 22], [3, 1019]]]

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 dependencies

Rule: RV_full_powerset_measurability_ORIGINAL_outcome_value_alignment · Uses: G001, S001

S003 · calculation

Before

RV ORIGINAL all aligned numerator weights := sum(1, 22, 3, 0)

After

RV ORIGINAL all aligned numerator weights = 26

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [[[0, -21, 1], [1, -21, 22], [2, 22, 3], [3, 1019, 0]], 26, -21]

After

derived = [26, [[0, -21, 1, Yes], [1, -21, 22, Yes], [2, 22, 3, No], [3, 1019, 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.

Step identifiers and dependencies

Rule: RV_point_mass_normalized_law_ON_S_and_ALL_fibre_terms · Uses: G001, S003

S005 · calculation

Before

RV ORIGINAL matching slot0 outcome0 mass := sum(0, 1)

After

RV ORIGINAL matching slot0 outcome0 mass = 1

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

RV ORIGINAL matching slot1 outcome1 mass := sum(1, 22)

After

RV ORIGINAL matching slot1 outcome1 mass = 23

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

RV ORIGINAL complete fibre mass over denominator := div(23, 26)

After

RV ORIGINAL complete fibre mass over denominator = 23/26

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [[[0, -21, 1], [1, -21, 22], [2, 22, 3], [3, 1019, 0]], -21, 23, 26]

After

derived = [23/26]

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 dependencies

Rule: RV_point_mass_unattained_value_zero_and_reduced_fraction · Uses: G001, S007

Reverse checks

  • S008 · 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.
  • S007 · premise_and_result_audit — Recompute RV ORIGINAL complete fibre mass over denominator with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute RV ORIGINAL matching slot1 outcome1 mass with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute RV ORIGINAL matching slot0 outcome0 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

23/26

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: c674aee911f0a740a6c77930c838aa3d238b0985a0e3ebded80c41f93f7ec7cb
  • candidate record SHA-256: a00d2a1577375f0db74370427e421d7bfe7be5f2bc40e766f0fa01a19e85067d
  • semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8
  • question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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