Monty Hall
Three doors. One hides a car, two hide goats. You pick door 1. The host — who knows what’s behind every door — opens door 3 to reveal a goat. They offer: stick with door 1, or switch to door 2. Does it matter?
It matters a lot. Switching wins two-thirds of the time.
Why switching wins
When you first pick, you have a 1/3 chance of being right. That means there’s a 2/3 chance the car is behind one of the other two doors. The host, forced to open a goat-door, collapses that 2/3 chance onto the single remaining unopened door.
- Stay: you win iff your first guess was right. Probability: 1/3.
- Switch: you win iff your first guess was wrong — because the host has eliminated the other wrong option for you. Probability: 2/3.
The host’s knowledge is doing all the work. If the host opened a random door (sometimes revealing the car themselves), the two strategies would be equally good. Information flows from their deliberate choice.
Why it feels wrong
Intuition says “two doors left, 50/50.” But the two remaining doors aren’t symmetric — one is the door you picked (1/3), the other is the door the host left closed (2/3). The asymmetry is subtle, which is why the result sparked a famous argument in the early 1990s.
Try
- Drag the trials slider down to 50 — the two curves bounce around and might even cross.
- Drag it to 10 000 — they converge cleanly to the theoretical rates.
- Resample a few times at n = 200. Notice how far either curve can wander early on; the 2/3 advantage is clear only at scale.