Mathematics
Optional Stopping Theorem and Fair Game Paradoxes
Quick fact
In a fair game, even if you use a clever stopping rule to 'quit while you're ahead,' your expected winnings are still exactly zero — you cannot outsmart a fair game. However, this only holds if your stopping rule has a bounded horizon or other technical conditions; otherwise, you can encounter paradoxes like the St. Petersburg lottery.
Why this is interesting
Imagine a perfectly fair coin flip where you win $1 on heads and lose $1 on tails. Surely you can't beat it... but what if you decide to stop after you're ahead by $1? Does that guarantee you walk away a winner?