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'''Feller's coin-tossing constants''' are a set of numerical constants which describe [[asymptotic]] [[probability|probabilities]] that in ''n'' independent tosses of a [[fair coin]], no run of ''k'' consecutive heads (or, equally, tails) appears. | |||
[[William Feller]] showed<ref>Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7</ref> that if this probability is written as ''p''(''n'',''k'') then | |||
:<math> | |||
\lim_{n\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k\, | |||
</math> | |||
where α<sub>''k''</sub> is the smallest positive real root of | |||
:<math>x^{k+1}=2^{k+1}(x-1)\,</math> | |||
and | |||
:<math>\beta_k={2-\alpha_k \over k+1-k\alpha_k}.</math> | |||
==Values of the constants== | |||
{|border=1 | |||
|- | |||
!k !!<math>\alpha_k</math> !!<math>\beta_k</math> | |||
|- | |||
|1||2||2 | |||
|- | |||
|2||1.23606797...||1.44721359... | |||
|- | |||
|3||1.08737802...||1.23683983... | |||
|- | |||
|4||1.03758012...||1.13268577... | |||
|} | |||
For <math>k=2</math> the constants are related to the [[golden ratio]] and [[Fibonacci numbers]]; the constants are <math>\sqrt{5}-1=2\varphi-2=2/\varphi</math> and <math>1-1/\sqrt{5}</math>. For higher values of <math>k</math> 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) = <math>\tfrac{9}{64}</math> = 0.140625. The approximation gives 1.44721356...×1.23606797...<sup>−11</sup> = 0.1406263... | |||
==References== | |||
{{Reflist}} | |||
==External links== | |||
* [http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Discrete_Structures/2200.aspx Steve Finch's constants at Mathsoft] {{broken link|date=November 2012}} | |||
[[Category:Mathematical constants]] | |||
[[Category:Games (probability)]] | |||
[[Category:Probability theorems]] | |||
Revision as of 23:14, 11 February 2013
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
where αk is the smallest positive real root of
and
Values of the constants
| k | ||
|---|---|---|
| 1 | 2 | 2 |
| 2 | 1.23606797... | 1.44721359... |
| 3 | 1.08737802... | 1.23683983... |
| 4 | 1.03758012... | 1.13268577... |
For the constants are related to the golden ratio and Fibonacci numbers; the constants are and . For higher values of 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) = = 0.140625. The approximation gives 1.44721356...×1.23606797...−11 = 0.1406263...
References
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External links
- ↑ Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7