>[!abstract] >The Monty Hall problem is a probability puzzle based on a game show scenario where a contestant chooses one of three doors: behind one is a prize, and behind the other two are goats. After the initial choice, the host (knowing the outcomes) opens one of the two unchosen doors to reveal a goat, then offers the contestant the chance to switch. Counterintuitively, the optimal strategy is to switch: the probability of winning by switching is 2/3, versus 1/3 by staying. The problem illustrates how conditional probability and information updating defy intuition, making it a classic example in probability theory and decision science. >[!note] Intuitive explanation >Imagine that there are 1,000,000 doors instead of just 3. You pick one at random. The host (who knows where the prize is) opens 999,998 doors, leaving just your door and (for example) door #735,872 unopened. It should now be obvious that the other door is far more likely to be the correct one (the odds go from 1/1,000,000 when picked at random, to 999,999/1,000,000 when picked by the host who knows which door leads to the prize is but won't open it). In essence, the host's question is really "do you want to keep your original door, or switch to *all other doors combined*?" The host is just adding a bit of showmanship by opening the non-selected door(s) before asking you the question. >[!related] >- **North** (upstream): [[Probability theory]] >- **West** (similar): [[Three prisoners problem]] >- **East** (different): [[Intuition-based reasoning]] >- **South** (downstream): [[Bayesian updating]]