Can You Rescue The Paratroopers?

A FiveThirtyEight Riddler puzzle.

By Vamshi Jandhyala in mathematics Riddler

February 23, 2018

Riddler Express

You have one token, and I have two tokens. Naturally, we both crave more tokens, so we play a game of skill that unfolds over a number of rounds in which the winner of each round gets to steal one token from the loser. The game itself ends when one of us is out of tokens — that person loses. Suppose that you’re better than me at this game and that you win each round two-thirds of the time and lose one-third of the time.

What is your probability of winning the game?

Solution

from random import random

runs = 100000
cnt = 0
for _ in range(runs):
    me, you = 1, 2
    while(you != 0 and me != 0):
        if random() <= 2/3:
            you -= 1
            me += 1
        else:
            you += 1
            me -= 1
        if you == 0:
            cnt += 1
print(cnt/runs) 

The probability of me winning the game is $.57$.