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

\lim _{n\rightarrow \infty }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}={2-\alpha _{k} \over 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, $\varphi$ , 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^[2] 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) = ${\tfrac {F_{n+2}^{(k)}}{2^{n}}}$ . ^[3]

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...×1.23606797...⁻¹¹ = 0.1406263...

References

↑ Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7
↑ Quora Question: "You toss a fair coin 12 times. What is the probability that two heads do not occur consecutively?"
↑ Coin Tossing at WolframMathWorld

External links

Steve Finch's constants at Mathsoft

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[1] Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7

[2] Quora Question: "You toss a fair coin 12 times. What is the probability that two heads do not occur consecutively?"

[3] Coin Tossing at WolframMathWorld