gistfile1.py

#!/usr/bin/env python3

import numpy as np

N = 22      # number of prisoners
M = 2       # number of switches

def state(n,m):
    # n is the number of uncounted prisoners
    # m is the value of the counter
    return (n << M) + m

P = np.identity(N<<M)

for n in range(N):
    for m in range(1<<M):
        s = state(n,m)
        if m < n and (m + 1) < (1<<M):
            t = state(n,m+1)
            p = (n - m)/N
            P[s,t] += p
            P[s,s] -= p
        t = state(n-m,0)
        p = 1/N
        P[s,t] += p
        P[s,s] -= p

# https://en.wikipedia.org/wiki/Absorbing_Markov_chain
R = np.matrix(np.identity(N<<M) - P)[1:, 1:,].getI()
print(R[state(N-1,0) - 1].sum())