Should You Pay $250 To Play This Casino Game?

Problem

Suppose a casino invents a new game that you must pay $250 to play. The game works like this: The casino draws random numbers between 0 and 1, from a uniform distribution. It adds them together until their sum is greater than 1, at which time it stops drawing new numbers. You get a payout of $100 each time a new number is drawn.

For example, suppose the casino draws 0.4 and then 0.7. Since the sum is greater than 1, it will stop after these two draws, and you receive $200. If instead it draws 0.2, 0.3, 0.3, and then 0.6, it will stop after the fourth draw and you will receive $400. Given the $250 entrance fee, should you play the game?

Specifically, what is the expected value of your winnings?

Solution

from random import random

runs = 10000

def average_winnings(payout, runs = runs):
    total = 0
    for _ in range(runs):
        total_draw = random()
        total += payout
        while(total_draw < 1):
            total_draw += random()
            total += payout
    return total/runs

print("Average winnings: ", average_winnings(100))

The average winnings is $\mathbf{272}$.