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 that if this probability is written as p(n,k) then
lim n → ∞ p ( n , k ) α k n + 1 = β k where αk is the smallest positive real root of
x k + 1 = 2 k + 1 ( x − 1 ) and
β k = 2 − α k k + 1 − k α k . For k = 2 the constants are related to the golden ratio, φ , and Fibonacci numbers; the constants are 5 − 1 = 2 φ − 2 = 2 / φ and 1 + 1 / 5 . The exact probability p(n,2) can be calculated either by using Fibonacci numbers, p(n,2) = F n + 2 2 n or by solving a direct recurrence relation leading to the same result. For higher values of k , 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) = F n + 2 ( k ) 2 n .
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) = 9 64 = 0.140625. The approximation p ( n , k ) ≈ β k / α k n + 1 gives 1.44721356...×1.23606797...−11 = 0.1406263...
