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 For
the constants are related to the golden ratio,
, and Fibonacci numbers; the constants are
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] 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) =