Feller's coin-tossing constants

Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails) appears. William Feller showed that if this probability is written as p(n,k) then lim n → ∞ p ( n , k ) α k n + 1 = β k {\displaystyle \lim _{n\rightarrow \infty }p(n,k)\alpha _{k}^{n+1}=\beta _{k}} where αk is the smallest positive real root of x k + 1 = 2 k + 1 ( x − 1 ) {\displaystyle x^{k+1}=2^{k+1}(x-1)} and β k = 2 − α k k + 1 − k α k .

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Feller's coin-tossing constants

Feller's coin-tossing constants are a set of numerical constants which describe asymptotic probabilities that in n independent tosses of a fair coin, no run of k consecutive heads (or, equally, tails) appears. William Feller showed that if this probability is written as p(n,k) then lim n → ∞ p ( n , k ) α k n + 1 = β k {\displaystyle \lim _{n\rightarrow \infty }p(n,k)\alpha _{k}^{n+1}=\beta _{k}} where αk is the smallest positive real root of x k + 1 = 2 k + 1 ( x − 1 ) {\displaystyle x^{k+1}=2^{k+1}(x-1)} and β k = 2 − α k k + 1 − k α k .

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Source: Wikipedia "Feller's coin-tossing constants" · CC BY-SA 4.0

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