December 11, 2012
by Bowen Kerins

My wife was recently playing the Cliff Hangers slot machine, based on a great game from The Price Is Right. Here’s how the bonus game works:

The goal is to climb exactly to the 25th and final step on the mountain (while yodeling, of course). You keep taking turns, picking from 5 doors, one each with the numbers 3, 5, 10, 15, and 20 on them. Whatever pick you make, that’s the number of steps taken on that turn. The doors are randomized, so you don’t know what you’re going to get, and you can get the same number more than once.

If after any number of turns your total is exactly 25 steps, you win. If you go over 25, you lose.

There are clearly some ways to win the game with lucky picking, such as 5 + 20 or 10 + 10 + 5. My wife won even though her first pick was a 3.

What is the probability of hitting 25 and winning this game? Which of these three goals gives the greatest chance of winning: 18, 55, or 58? If there were no stopping point, what is the long-term probability of hitting a high number like 507?

We’ll provide a solution next week that involves a single mathematical calculation. It’s pretty awesome.