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 [Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7] that if this probability is written as "p"("n","k") then

:

where α_"k" is the smallest positive real root of

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

and

:

Values of the constants

For $k=2$ the constants are related to the golden ratio and Fibonacci numbers; the constants are $sqrt\{5\}-1=2varphi-2=2/varphi$ and $1-1/sqrt\{5\}$. For higher values of $k$ they are related to generalizations of Fibonacci numbers such as the tribonacci and tetranacci constants.

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 gives 1.44721356...×1.23606797...⁻¹¹ = 0.1406263...

References

External links

* [http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Discrete_Structures/2200.aspx Steve Finch's constants at Mathsoft]