The game-show paradox that fools almost everyone, Ph.D.s included. You pick one of three doors; behind one is a car, behind the others, goats. The host — who knows what's where — opens a different door to reveal a goat, then offers to let you switch to the last closed door. It feels like a 50/50 toss-up. It isn't: switching wins twice as oftenYour first pick is right 1/3 of the time and stays 1/3 — opening a known goat elsewhere can't change a guess you already made. The other 2/3 collapses onto the single door you can switch to. as staying.
Switching wins 2× as often as staying. Your first pick is right just 33.3% of the time, and all the rest of the odds — 66.7% — pile onto the one door left closed.
Each line is the running share of games that strategy has won so far. They lurch around early — a handful of games is mostly luck — then settle firmly onto the dashed theoretical odds as the count climbs. Seeded from the inputs, so this exact run reproduces for anyone who opens the link.
| Doors (host opens all but one) | Stay | Switch |
|---|---|---|
| 3 doorsclassicyou | 33.3% | 66.7% |
| 5 doors | 20% | 80% |
| 10 doors | 10% | 90% |
| 25 doors | 4% | 96% |
| 100 doors | 1% | 99% |
Crank the doors up and the intuition finally clicks. With 100 doors you pick one — a 1-in-100 shot — and the host throws open 98 goats, leaving your door and one other. The car is almost certainly behind that other door, because your blind first guess almost certainly missed. Switching is a near-sure thing; staying is the original long shot, untouched.