# Question 15012: Evaluate the Deviation Profile Only at a Valid Probability Index

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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.

R0029-15012 · q-20261014-advance-20261030-r07-convergence-in-probability-0012



## Original question and givens

Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=12. Give (numerator,shift,n,deviation_probability).


- G001 — Use the declared rational deviation-probability tail p_n=2/(n+6) on nonnegative integer indices where this value belongs to[0,1], starting no earlier than max(0,numerator-shift). The common probability space and infinite-process interpretation remain declared premises, not constructions inferred from finitely many numbers. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Evaluate index n=12. Give (numerator,shift,n,deviation_probability).





## 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-convergence-in-probability](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/)





## Step-by-step derivation

S001 · impliesBefore

positive profile parameters c=2;s=6;raw fraction=c/(n+s)

After

nonnegative valid probability indices start at=0;only there0<profile<=1

Conditions: none

A positive denominator and value≤1 require n+s≥c.Keep the original index;do not clip values above1 or silently shift them.This checks arithmetic domain only.The infinite-process/common-space interpretation remains declared,and the prefix-only family gains no infinite continuation premise.

Step identifiers and dependenciesRule: deviation_profile_probability_domain_not_clipping · Uses: G001





S002 · calculationBefore

deviation denominator at original index := sum(12, 6)

After

deviation denominator at original index = 18

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

deviation probability at original index := div(2, 18)

After

deviation probability at original index = 1/9

Conditions: none

Compute exactly from original roles and established intermediates,retaining denominators,strict guards,indices and signs.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Recompute deviation probability at original index with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute deviation denominator at original index with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.





## Result

(2, 6, 12, 1/9)

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: 9b4da8183649ad982877262b1e7ea020ef2f94de482cc94ff626fa54a1db6f4b
- candidate record SHA-256: aff1415ffd75130b4dde98dde54a3407a361290826f7f7e14753196a8ef7960e
- semantic review receipt SHA-256: 8c0ef6ca8c5ba2754e41f30c2b32bcb757756152d97421f7396d39038c11f725
- question-first witness SHA-256: c7920a0ff8e724b7e566ebbda30b31d8087c136974b62f8c760fbd26ab816117
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261030-r07-convergence-in-probability-0012.json)