see https://www.greaterwrong.com/posts/gptXmhJxFiEwuPN98/meetup-notes-ole-peters-on-ergodicity

stpetersburg.py

#!/usr/bin/env python3

import numpy as np

def flip():
    return np.random.choice([0, 1])

def play(wealth, cost, max_rounds=1e5):
    num_rounds = 0
    while wealth > cost and num_rounds < max_rounds:
        round_winnings = 1
        while True:
            coin = flip()
            if coin == 0:
                round_winnings *= 2
            else:
                break
        wealth = wealth - cost + round_winnings
        num_rounds += 1
    return {"wealth": wealth, "num_rounds": num_rounds}

wealth = 1000  # starting wealth
cost = 9  # cost per round to play the game
print("Using starting wealth of {} and cost per round of {} below.\n".format(wealth, cost))

# first, play for as long as possible
print("Here are some typical rounds, playing for as long as possible:\n")
for _ in range(5):
    result = play(wealth, cost)
    print("* survived {} rounds with remaining wealth {}".format(result["num_rounds"], result["wealth"]))

# next, commit to playing at most r rounds, and find probability of leaving
# with more money than you started with
print("\nNow we will commit to play a fixed number of rounds (or until\n"
      "wealth runs out), and find the probability that we leave with\n"
      "more money than we start with:\n")
for r in [10, 100, 1000, 10000]:
    wins = 0
    losses = 0
    for _ in range(1000):
        if play(wealth, cost, max_rounds=r)["wealth"] > wealth:
            wins += 1
        else:
            losses += 1
    print("* committing to play {} rounds: we had {} wins and {} losses, so {}% chance of winning".format(r, wins, losses, round(wins/(wins + losses)*100, 1)))

Typical output of the program:

Using starting wealth of 1000 and cost per round of 9 below.

Here are some typical rounds, playing for as long as possible:

• survived 171 rounds with remaining wealth 3
• survived 798 rounds with remaining wealth 7
• survived 1460 rounds with remaining wealth 4
• survived 163 rounds with remaining wealth 4
• survived 717 rounds with remaining wealth 3

Now we will commit to play a fixed number of rounds (or until
wealth runs out), and find the probability that we leave with
more money than we start with:

• committing to play 10 rounds: we had 109 wins and 891 losses, so 10.9% chance of winning
• committing to play 100 rounds: we had 179 wins and 821 losses, so 17.9% chance of winning
• committing to play 1000 rounds: we had 83 wins and 917 losses, so 8.3% chance of winning
• committing to play 10000 rounds: we had 17 wins and 983 losses, so 1.7% chance of winning