Mathematics
Martingales and the Optional Stopping Theorem
Quick fact
A symmetric random walk, where you gain or lose one unit with equal probability, is a martingale. Yet the first time it hits either +100 or -100, the expected value is 0, matching the starting value—even though you stop at a random time that depends on the process itself.
Why this is interesting
Imagine you are at a casino with an unbeatable betting system. No matter how clever you are, the casino always wins in the long run—but why? The answer lies in a deceptively simple property called the martingale.