Question 14421: Exact Bernoulli Cylinder Probability under a Declared Product Law

Original question and givens

Use the explicitly declared Bernoulli product-law model with exact p=21/29 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, 1, 0]. Give (word,ones,zeros,p,probability,symbolic_product_law).

• G001 — Use the explicitly declared Bernoulli product-law model with exact p=21/29 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, 1, 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

Step-by-step derivation

Bernoulli zero-outcome mass := sub(1, 21/29)

Bernoulli zero-outcome mass = 8/29

raw Bernoulli parameter=[21, 29]

normalized p=21/29;mass at0=8/29;mass at1=21/29;symbolic product-law model explicitly declared

original ordered finite binary word=[0, 0, 1, 0];p=21/29;one-minus-p=8/29

cylinder atom factorization uses the same declared symbolic product law

binary word number of ones := sum(0, 0, 1, 0)

binary word number of ones = 1

binary word number of zeros := sub(4, 1)

binary word number of zeros = 3

(21/29)^1

Bernoulli one factors power = 21/29

(8/29)^3

Bernoulli zero factors power = 512/24389

Bernoulli cylinder product := mul(21/29, 512/24389)

Bernoulli cylinder product = 10752/707281

word length=4;ones=1;zeros=3;cylinder probability=10752/707281

arbitrary-word formula p^ones*(1-p)^zeros;empty product1 including degenerate parameters

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, 1, 0], 1, 3, 21/29, 10752/707281, 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.