# Question 14417: Exact Bernoulli Cylinder Probability under a Declared Product Law [Return to catalogue page 73](https://ocme.evemisslab.com/en/questions/page/73/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261029-r06-iid-sequence-foundations/#q-20261014-advance-20261029-r06-iid-sequence-0017) 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. R0027-14417 · q-20261014-advance-20261029-r06-iid-sequence-0017 ## Original question and givens Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law). - G001 — Use the explicitly declared Bernoulli product-law model with exact p=17/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. Ordered finite binary word=[0, 0, 0, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law). ## 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-iid-sequence](https://ocme.evemisslab.com/en/mko/mko-iid-sequence/) ## Step-by-step derivation S001 · calculationBefore Bernoulli zero-outcome mass := sub(1, 17/29) After Bernoulli zero-outcome mass = 12/29 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · impliesBefore raw Bernoulli parameter=[17, 29] After normalized p=17/29;mass at0=12/29;mass at1=17/29;symbolic product-law model explicitly declared Conditions: none Retain the complete{0,1} marginal and the explicitly declared product-law premise.Infinite IID is not inferred from a data set,equal means or pairwise checks. Step identifiers and dependenciesRule: Bernoulli_complete_marginal_and_product_law_premise · Uses: G001, S001 S003 · impliesBefore original ordered finite binary word=[0, 0, 0, 0];p=17/29;one-minus-p=12/29 After cylinder atom factorization uses the same declared symbolic product law Conditions: none Preserve word order,use p for1 and1−p for0.Factorization is a model premise,not a test proving independence of an unknown dependent law.Retain empty words and p=0 or1 boundaries. Step identifiers and dependenciesRule: IID_ordered_word_and_given_model_not_unknown_law_test · Uses: G001, S002 S004 · calculationBefore binary word number of ones := sum(0, 0, 0, 0) After binary word number of ones = 0 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore binary word number of zeros := sub(4, 0) After binary word number of zeros = 4 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore (17/29)^0 After Bernoulli one factors power = 1 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S005 S007 · impliesBefore (12/29)^4 After Bernoulli zero factors power = 20736/707281 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S006 S008 · calculationBefore Bernoulli cylinder product := mul(1, 20736/707281) After Bernoulli cylinder product = 20736/707281 Conditions: none Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · impliesBefore word length=4;ones=0;zeros=4;cylinder probability=20736/707281 After arbitrary-word formula p^ones*(1-p)^zeros;empty product1 including degenerate parameters Conditions: none Collect identical marginal factors by counts without sorting or changing the original word.Exponent0 denotes empty-product1,including p=0 or1.This finite evaluation is not a new proof of IID for an unknown infinite sequence. Step identifiers and dependenciesRule: Bernoulli_cylinder_formula_empty_product_boundary · Uses: G001, S008 ## Reverse checks - 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 Bernoulli cylinder product 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute binary word number of zeros with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute binary word number of ones with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals. ## Result ([0, 0, 0, 0], 0, 4, 17/29, 20736/707281, symbolic_product_law) 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: a3085254068cd5c313dbfb768030808933f7bd79dfc2d13fde4aa6ce9ac6c84f - candidate record SHA-256: 1b36fec2616dac3674d1113c9203e8b8cca82214214402a1dee200c8abc82c92 - semantic review receipt SHA-256: e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe - question-first witness SHA-256: a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261029-r06-iid-sequence-0017.json)