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^[1] that if this probability is written as p(n,k) then

${\displaystyle \lim _{n\rightarrow \infty }p(n,k)\alpha _{k}^{n+1}=\beta _{k}}$

where α_k is the smallest positive real root of

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

and

${\displaystyle \beta _{k}={2-\alpha _{k} \over k+1-k\alpha _{k}}.}$

Values of the constants

k	${\displaystyle \alpha _{k}}$	${\displaystyle \beta _{k}}$
1	2	2
2	1.23606797...	1.44721359...
3	1.08737802...	1.23683983...
4	1.03758012...	1.13268577...

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

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) = ${\displaystyle {\tfrac {9}{64}}}$ = 0.140625. The approximation ${\displaystyle p(n,k)\approx \beta _{k}/\alpha _{k}^{n+1}}$ gives 1.44721356...×1.23606797...⁻¹¹ = 0.1406263...