I’ve thought of a gambling problem (read: probability problem) that I can’t seem to solve using the methods I know of. The problem is to find an optimal playing strategy for the following game:
Pay me $100. I give you one chip (worth $1). You can decide to keep your $1 and walk away (down $99) or flip a coin with me. If it comes up heads, then you have three chips. If it comes up tails, you walk away with nothing ($100 down). If you have three chips, then you can walk away with $3 ($97 down), or try for nine chips. And so on.
The thing that makes this game hard to analyze is its infinite expectation. Every time you go for the coin, your payback is greater than your odds. For instance (ignore the $100 for now): I have one chip. I can walk away with $1. Or I can flip and get 0 1/2 of the time, and 3 1/2 of the time, for an expectation of 1.5 chips. Clearly I should flip. The exact same reasoning happens at every stage of the game. You hit every time, and you end up losing it all, because the coin is going to come up tails eventually.
It has something to do with that $100 for one chip (Keep in mind that hat’s the only way you can play this game). Because without it, the game is pretty easy to analyze. Give me some money, and I’ll triple it with 1/2 probability, or take it all. To play that in real life, you just play with amounts of money such that if you lose, you can play again. Play with $10. Get $30. Play with $10 again. Lose it. Keep playing like that and you can get as rich as you want.