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The St. Petersburg Paradox
There's no upper limit to the amount you could win here, which would suggest you ought to be willing to pay almost any quantity of money. The question is what is a fair price to pay to be allowed to play this game? And this flummox probability theorist for a very long time because if you do the maths, your expected possible gain is infinite. It didn't seem possible it was random. How many people need to be in a room to have the same birthday? And why does it matter? Twenty-three is the answer.