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 constantsEdit

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  . 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]

ExampleEdit

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...

ReferencesEdit

  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

External linksEdit