Monty Hall

The paradox, briefly

After you pick a door, Monty — who knows what's behind all three — always opens one of the other two to show a goat. That's not a random act; it's a deliberate move that dumps the probability the two unpicked doors used to share onto whichever one he leaves closed. Your original pick is still stuck at 1-in-3. The door he left standing now carries the other 2-in-3. So switching wins twice as often as sticking — even though, once only two doors remain, it feels like it should be a coin flip.

Why it's worth playing, not just reading

Almost everyone's gut says 50/50 at the two-door stage — that misfire is the entire point of the puzzle. It isolates one specific, very common reasoning error (treating a filtered choice as a fresh coin flip) and lets you catch yourself making it in real time, instead of taking someone else's word for the answer.

Why it's fun

Each round is a quick bet against your own instinct, with an instant, visual payoff. Auto-simulate turns that single bet into evidence: run a few thousand trials and watch the bars visibly settle onto the 1/3 and 2/3 lines in about a second — the law of large numbers made tangible instead of asserted.

Why it helps learning

It's a compact, self-contained demo of conditional probability and Bayesian updating — the same reasoning behind medical test results, spam filters, and weighing evidence. Get an intuition for it here with doors and goats, and it transfers directly to reasoning you'll actually use elsewhere.