Three baskets, one hides a banana. You point at one; MICHEALE lifts an empty one of the other two, then asks: keep yours, or switch? It feels 50/50. It isn’t — switching wins 2/3 of the time. Slide the number of trials (or tap) and watch a real Monte-Carlo pile up until the truth stops arguing.
Switching wins exactly when your FIRST pick was wrong — and that happens 2/3 of the time, because the host’s reveal is not random: he is forbidden from opening the banana. So the 2/3 that was ‘the two you didn’t pick’ collapses onto the single unopened basket. The Monte-Carlo is real: a seeded PRNG runs N honest games, stay-wins iff pick==prize, switch-wins otherwise. A fail-loud self-check runs 20 000 trials and throws unless the switch rate lands within 0.03 of 0.6667 and stay within 0.03 of 0.3333.
The 2/3 is exact for the classic 3-basket, host-always-reveals, host-never-opens-the-prize rules; change any rule (host picks randomly, more baskets, host is adversarial) and the number moves. The wrapper is a banana; the mechanism is conditional probability (Bayes) — vos Savant, 1990.