# Question 20300: The Exact Tail Is a Sum over the Full Threshold Event [Return to catalogue page 102](https://ocme.evemisslab.com/en/questions/page/102/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261107-r04-markov-inequality-foundations/#q-20261014-advance-20261107-r04-markov-inequality-0050) 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. R0039-20300 · q-20261014-advance-20261107-r04-markov-inequality-0050 ## Original question and givens Use complete normalized finite law with exact masses=[1/30, 2/30, 27/30] and aligned total values X=[0, 25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=26. Give (threshold, tail_probability, certificate). - G001 — Use complete normalized finite law with exact masses=[1/30, 2/30, 27/30] and aligned total values X=[0, 25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=26. Give (threshold, tail_probability, certificate). ## 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-markov-inequality](https://ocme.evemisslab.com/en/mko/mko-markov-inequality/) ## Step-by-step derivation S001 · impliesBefore source operation=markov-tail-probability;original snapshot and mathematics retained After explicit ordered source output fields=[threshold, tail_probability, certificate] Conditions: none Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Markov complete probability mass := sum(1/30, 1/15, 9/10) After Markov complete probability mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Markov original atom0 weighted X := mul(1/30, 0) After Markov original atom0 weighted X = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Markov original atom1 weighted X := mul(1/15, 25) After Markov original atom1 weighted X = 5/3 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Markov original atom2 weighted X := mul(9/10, 27) After Markov original atom2 weighted X = 243/10 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Markov complete SAME-law expectation := sum(0, 5/3, 243/10) After Markov complete SAME-law expectation = 779/30 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore ORIGINAL complete nonempty masses=[1/30, 1/15, 9/10];ALL aligned values X=[0, 25, 27];full mass=1;ALL weighted X terms=[0, 5/3, 243/10] After finite exact expectation=779/30;prototype EVERY-ENTRY nonnegative=Yes;actual positive-mass nonnegativity=Yes;null-negative atoms=[];positive-mass negative atoms=[] Conditions: none The prototype flag tests every table entry,whereas almost-sure nonnegativity concerns positive-mass atoms only.The English source-review guard excludes null-only negative ambiguities from non-tail families rather than calling them a theorem-premise failure.The tail itself can still be computed for signed X. Step identifiers and dependenciesRule: markov_pointwise_source_gate_not_almost_sure_confusion · Uses: G001, S006 S008 · calculationBefore Markov original atom0 threshold-event mass := mul(1/30, 0) After Markov original atom0 threshold-event mass = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore ORIGINAL atom=0;value X=0;positive threshold=26;original mass=1/30 After event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0 Conditions: none The event is X≥threshold,not X>threshold.Multiply by the event indicator at every original atom,including null atoms;use the complete law rather than observed frequencies. Step identifiers and dependenciesRule: markov_full_tail_event_includes_equality_not_strict_tail · Uses: G001, S008 S010 · calculationBefore Markov original atom1 threshold-event mass := mul(1/15, 0) After Markov original atom1 threshold-event mass = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore ORIGINAL atom=1;value X=25;positive threshold=26;original mass=1/15 After event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0 Conditions: none The event is X≥threshold,not X>threshold.Multiply by the event indicator at every original atom,including null atoms;use the complete law rather than observed frequencies. Step identifiers and dependenciesRule: markov_full_tail_event_includes_equality_not_strict_tail · Uses: G001, S010 S012 · calculationBefore Markov original atom2 threshold-event mass := mul(9/10, 1) After Markov original atom2 threshold-event mass = 9/10 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore ORIGINAL atom=2;value X=27;positive threshold=26;original mass=9/10 After event X>=threshold=Yes;EQUALITY INCLUDED;event mass contribution=9/10 Conditions: none The event is X≥threshold,not X>threshold.Multiply by the event indicator at every original atom,including null atoms;use the complete law rather than observed frequencies. Step identifiers and dependenciesRule: markov_full_tail_event_includes_equality_not_strict_tail · Uses: G001, S012 S014 · calculationBefore Markov complete exact tail probability := sum(0, 0, 9/10) After Markov complete exact tail probability = 9/10 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore ALL original threshold-event flags=[No, No, Yes];ALL law-weighted indicator terms=[0, 0, 9/10] After exact complete-law tail probability=9/10;this event calculation ALONE does not assert Markov applicability Conditions: none The complete event-mass sum is the actual tail probability,even for signed X.That calculation neither proves almost-sure nonnegativity nor identifies the tail with the later expectation-over-threshold bound. Step identifiers and dependenciesRule: markov_exact_event_probability_separate_from_theorem_bound · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Recompute Markov complete exact tail probability with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute Markov original atom2 threshold-event mass with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute Markov original atom1 threshold-event mass with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Recompute Markov original atom0 threshold-event mass with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute Markov complete SAME-law expectation with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Markov original atom2 weighted X with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Markov original atom1 weighted X with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Markov original atom0 weighted X with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Markov complete probability mass 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 (26, 9/10, exact_complete_finite_tail) 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: 845b1de07c8edd75126b36977183a722f1089b75395a1917007b3a8c50d74963 - candidate record SHA-256: 5ef3ce26c1b9fcead98730e7966799bb607b164bb95494dda98a74a9256249d2 - semantic review receipt SHA-256: 6e11e35fafe32f16900ce9b522d01acb7b2ee6b463ab179392076cfbe1af8034 - question-first witness SHA-256: cc0fa361fc25151729e70264340d2eae94f179f6f2a2edfe80ba5704fbbd0682 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261107-r04-markov-inequality-0050.json)