Optional stopping theorem

The Optional Stopping Theorem states that, under certain conditions, the expected value of a fair game or process at a stopping point (when you decide to stop based on information up to that point) is the same as its initial expected value. In simpler terms, if you're betting in a fair game and decide when to stop based on what you've observed, your expected winnings should remain unchanged. This principle helps ensure that, with proper rules and timing, stopping doesn't give you an unfair advantage or lead to expected gains or losses over time.