Английская Википедия:Feller's coin-tossing constants

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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

<math>

\lim_{n\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k </math>

where α_k is the smallest positive real root of

<math>x^{k+1}=2^{k+1}(x-1)</math>

and

<math>\beta_k={2-\alpha_k \over k+1-k\alpha_k}.</math>

Values of the constants

k	<math>\alpha_k</math>	<math>\beta_k</math>
1	2	2
2	1.23606797...	1.44721359...
3	1.08737802...	1.23683983...
4	1.03758012...	1.13268577...

For <math>k=2</math> the constants are related to the golden ratio, <math>\varphi</math>, and Fibonacci numbers; the constants are <math>\sqrt{5}-1=2\varphi-2=2/\varphi</math> and <math>1+1/\sqrt{5}</math>. The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = <math>\tfrac{F_{n+2}}{2^n}</math> or by solving a direct recurrence relation leading to the same result. For higher values of <math>k</math>, the constants are related to generalizations of Fibonacci numbers such as the tribonacci and tetranacci numbers. The corresponding exact probabilities can be calculated as p(n,k) = <math>\tfrac{F^{(k)}_{n+2}}{2^n}</math>.

Example

If we toss a fair coin ten times then the exact probability that no pair of heads come up in succession (i.e. n = 10 and k = 2) is p(10,2) = <math>\tfrac{9}{64}</math> = 0.140625. The approximation <math>p(n,k) \approx \beta_k / \alpha_k^{n+1}</math> gives 1.44721356...×1.23606797...⁻¹¹ = 0.1406263...

External links

• Steve Finch's constants at Mathsoft