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

where αk is the smallest positive real root of

and

Values of the constants

k
122
21.23606797...1.44721359...
31.08737802...1.23683983...
41.03758012...1.13268577...

For the constants are related to the golden ratio, , and Fibonacci numbers; the constants are and . The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) =  or by solving a direct recurrence relation leading to the same result. For higher values of , 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) = . [2]

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

  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. Coin Tossing at WolframMathWorld
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.