The OST says that given a finite amount of money and a finite time horizon, any betting sequence/strategy cannot improve the expected earnings from a sequence of fair gambles. More to the point, the theorem is proof that you can’t expect to make money playing a fair game (a martingale) and you can’t bet in such a way that you can expect to beat an unfair game (a supermartingale). Attached, I’ve written a rundown of the result along with a nice short proof.

The Optional Stopping/Sampling Theorem

Stochastic Process, Optional Stopping Theorem, Expected Value, Conditional Expectation, Martingale