little problem(s) 2022.09

[Albert Chan]

Recursive formula for obtaining k consecutive heads from n coin tosses

50 coin tosses, runs of 5 tails or more. Probability of tails = 1 - 0.6 = 0.4

>>> k, N = 5, 50
>>> p, P = 0.4, [0]*(N+1)
>>> P[k] = p**k
>>> for n in range(k+1,N+1): P[n] = P[n-1] + (1-P[n-k-1])*(1-p)*p**k
...
>>> P[N]
0.25645932207797717

Confirmed by simulations:

>>> from random import random
>>> pat = lambda: ''.join("HT"[random()<=p] for x in range(N))
>>> sum('TTTTT' in pat() for i in range(10**6))
256541

Real life is clumpy
The inherent clumpiness of randomness
The Longest Run of Heads - Mark F. Schilling