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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.
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| [[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 | |
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| :<math>
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| \lim_{n\rightarrow \infty} p(n,k) \alpha_k^{n+1}=\beta_k\,
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| </math>
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| where α<sub>''k''</sub> is the smallest positive real root of
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| :<math>x^{k+1}=2^{k+1}(x-1)\,</math>
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| and
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| :<math>\beta_k={2-\alpha_k \over k+1-k\alpha_k}.</math>
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| ==Values of the constants==
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| {|border=1
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| |-
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| !k !!<math>\alpha_k</math> !!<math>\beta_k</math>
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| |1||2||2
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| |-
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| |2||1.23606797...||1.44721359...
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| |-
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| |3||1.08737802...||1.23683983...
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| |-
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| |4||1.03758012...||1.13268577...
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| |}
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| 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. | |
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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) = <math>\tfrac{9}{64}</math> = 0.140625. The approximation gives 1.44721356...×1.23606797...<sup>−11</sup> = 0.1406263...
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/Discrete_Structures/2200.aspx Steve Finch's constants at Mathsoft] {{broken link|date=November 2012}}
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| [[Category:Mathematical constants]]
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| [[Category:Games (probability)]]
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| [[Category:Probability theorems]]
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