(This was the “puzzler” on 538.com recently)

There’s a game at a fictional casino. If you choose to play, the casino randomly generates a number between 0 and 1. Then they generate another number between 0 and 1, and add it to the first. They repeated this process until the sum of all the generated numbers is greater than 1.

They then pay you $100 for every random number they generated.

So if the random numbers were .25, then .42, then .61, you would be paid $300, because .25 + .42 is not greater than 1, but adding in .61 is, and they generated three numbers total.

They charge you $250 to play this game.

Is that a good deal? Why or why not? And how much should you be willing to pay to play?

My (partial) answer:

I’m quite certain that you should pay $250 to play the game.

Obviously you always get at least 2 pulls, so at least $200. The question is how often does it take at least 3 pulls, so you’re coming out ahead, with $300.

So let’s restrict the casino real numbers to fractions with n as the denominator, so from 0 through 1/n, 2/n, up to n/n=1. What are the odds that two random such fractions sum to 1 or less? Well, if the first number they chose was 0, then it’s certain, n/n. If the first number they chose was 1/n, then it’s (n-1)/n. If the first number they chose was 2/n, then it’s (n-2)/n. Each of their possible choices was equally likely for their first pick, so the sum of all those probabilities ends up being 1 + 2 + 3 … + n/n^2, which, as n approaches infinity, becomes 1/2. So if you maxed out at 3 flips, you would still get 2 flips half the time and 3 flips half the time, so that’s break even at $250. Since you have some non-zero chance of winning more than $300, the game is definitely in your favor.

As for what the actual EV is, I have absolutely no idea.

Some puzzles like this the answer is that the EV is infinite. After all, there’s a non-zero probability that you win a googol dollars. But I don’t think that’s the case here, because as you get to larger and larger amounts, the value you win increases linearly but the likelihood of winning it increases exponentially.

That’s all I got… any of our math experts want to take a crack at this? (Or is it a well known problem?)