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

:$\lim_ p(n,k) \alpha_k^=\beta_k$

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

:$x^=2^(x-1)$

and

:$\beta_k=.$

Values of the constants

For $k=2$ the constants are related to the golden ratio, $\varphi$, and Fibonacci numbers; the constants are $\sqrt-1=2\varphi-2=2/\varphi$ and $1+1/\sqrt$. The exact probability ''p''(n,2) can be calculated either by using Fibonacci numbers, ''p''(n,2) = $\tfrac$ 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) = $\tfrac$. Coin Tossing at WolframMathWorld
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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) = $\tfrac$ = 0.140625. The approximation $p(n,k) \approx \beta_k / \alpha_k^$ gives 1.44721356...×1.23606797...⁻¹¹ = 0.1406263...

References

{{Reflist

External links

Steve Finch's constants at Mathsoft

Mathematical constants
Gambling mathematics
Probability theorems