The Monty Hall Problem

Three doors stand in a row. Behind one, a new car. Behind the other two, goats. You pick door number one. The host — and this matters — knows where the car is. He walks to door number three and opens it. A goat. He turns to you and asks: do you want to switch to door number two? Your gut says it does not matter. Two doors left, one car. Fifty-fifty. Stay or switch, same odds. That feeling is strong, and it is wrong. Before the host did anything, you picked one door out of three. The chance you picked the car: one in three. The chance the car is behind one of the other two doors: two in three. Nothing controversial yet. One third for your door, two thirds for the other side. Now the host opens a door from that other side — but here is the detail that changes everything. He does not open a door at random. He knows where the car is, and he always opens a door with a goat.