St. Petersburg Paradox
Imagine a game where a fair coin is tossed repeatedly until ‘Heads’ appears. If it appears on the 1st toss, you win €2. If it takes until the 2nd toss, you win €4. If it takes until the 3rd toss, you win €8, doubling every time.
Mathematically speaking, what is the fair price you should be willing to pay to play this game just once if you want to break even?
If you calculate the expected value ($E$) of the game using pure probability, the math looks like this:
$$E = \left(\frac{1}{2} \times 2\right) + \left(\frac{1}{4} \times 4\right) + \left(\frac{1}{8} \times 8\right) + \dots = 1 + 1 + 1 + \dots = \infty$$Because the prize keeps doubling at the exact same rate that the probability of winning halves, the mathematical expectation is literally infinite. Therefore, from a purely theoretical standpoint, you should be willing to wager everything you own to play it just once.
The Paradox: No sane person would actually pay more than a few euros to play this game. This happens because real-world casinos don’t have infinite bankrolls to pay out if you flip a long streak of tails, and human beings naturally maximize expected utility (the psychological value of money) rather than pure expected financial value.