Ever wondered why betting on a roulette wheel doesn't change your odds over time?
Image: Internet Archive Book Images, No restrictions, via Wikimedia Commons
the optional stopping theorem says about martingales and stopping times
Ever wondered why betting on a roulette wheel doesn't change your odds over time?
Imagine playing a game where you bet $1 on a roulette wheel. After each spin, you get back your bet plus your winnings. You keep betting until you either win or lose all your money.
It's like you're in a video game where your character's health doesn't decrease or increase over time, just fluctuates randomly. The concept we're talking about is a "martingale," which is a mathematical model for this type of game.
Example
After 10 spins, you might have lost 5. You bet 1 again, hoping to recover your loss. But statistically, if you keep playing, your average loss per game will eventually balance out to zero over time.
Remember this
The key insight is that in a fair game modeled by a martingale, your expected winnings don't change over time, even though individual outcomes can vary widely.
Text adapted from Wikipedia, licensed under CC BY-SA 4.0.
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