Hitting Time of Coin

# Given a coin probability p, what is the expected hitting time of getting k trials in a row?
p = 0.505
k = 14

# Analytical solution
failure_p = sum(p ** i * (1 - p) for i in range(k))
cycle_time = sum(p ** i * (1 - p) * (i + 1) for i in range(k))
expected_cycles = 1 / (1 - failure_p)
print(cycle_time * expected_cycles + k)

# Stochastic verification
import random
trials = 100
average_n = 0
for i in range(trials):
    in_a_row = 0
    n = 0
    while in_a_row != k:
        n += 1
        if random.random() < p:
            in_a_row += 1
        else:
            in_a_row = 0
    average_n += n
print(average_n / trials)