Here's a fun puzzle from the same guys that introduced me to the infinite spheres of utility problem. This one's more along the lines of the Doomsday argument, or perhaps the Sleeping Beauty paradox. Anyway, here's the set up: a group of people are sent into a room. Two dice are rolled, and if they land double sixes then everyone in the room gets shot. Otherwise, they're released and the whole procedure is repeated again with a new group of ten times as many people. (And so on, until a group gets shot.) You're sent into the room. What is your chance of being shot?
Well, obviously 1/36, right? That's the chance of double sixes being rolled. But note that the vast majority (~90%) of people who enter the room get shot. (This is stipulated -- assume there is an unlimited stock of people, ammo, etc. There's no risk of "running out", no matter how many rounds the game goes on for.) You have no reason to consider yourself one of the lucky 10%. You have the same chance of being shot that anyone else who enters the room does. So you must all have a 9/10 chance of being shot.
Very puzzling. I'm inclined to insist that in fact each person only has a 1/36 chance of being shot, even though 90% of the people will be shot. Chance is about causal bases (or something), not frequencies.
Curiously, this means that an immortal with an indefinite overdraft could make a lot of money by betting with every single individual about to enter the room. But then, an immortal with an indefinite overdraft can strategically gamble and profit from almost any repeated game, no matter how poor the odds. (Just keep doubling your bet until you win, then pocket the winnings and start over.) So maybe the same kind of thing is going on here.