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.
where αk is the smallest positive real root of
Values of the constants
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.
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...
- Feller, W. (1968) An Introduction to Probability Theory and Its Applications, Volume 1 (3rd Edition), Wiley. ISBN 0-471-25708-7 Section XIII.7