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

$\underset{n\to \mathrm{\infty }}{\lim }p(n,k){\alpha }_k^{n+1}={\beta }_k$

where α_k is the smallest positive real root of

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

and

${\beta }_k=\frac{2-{\alpha }_k}{k+1-k{\alpha }_k}.$

Values of the constants

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

For $k=2$ the constants are related to the golden ratio, $\phi$, and Fibonacci numbers; the constants are $\sqrt{5}-1=2\phi -2=2/\phi$ and $1+1/\sqrt{5}$. The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = $\frac{F_{n+2}}{2^n}$ 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) = $\frac{F_{n+2}^{(k)}}{2^n}$.

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

External links

• Steve Finch's constants at Mathsoft