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\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k $$

where &alpha;k is the smallest positive real root of


 * $$x^{k+1}=2^{k+1}(x-1)$$

and


 * $$\beta_k={2-\alpha_k \over k+1-k\alpha_k}.$$

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

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) = $$\tfrac{9}{64}$$ = 0.140625. The approximation $$p(n,k) \approx \beta_k / \alpha_k^{n+1}$$ gives 1.44721356...&times;1.23606797...&minus;11 = 0.1406263...