The Monty Hall problem

The problem
123

You're on a game show. There are three doors. Behind one is a car, behind the other two are goats. You pick door 1. The host — who knows what's behind every door — opens door 3 to reveal a goat. Then he asks: do you want to switch to door 2, or stay with door 1?

Tempting (but wrong)
123??50 / 50 ?

It feels obvious. Two doors are left. One has a car, the other has a goat. So it's 50/50, and it doesn't matter whether you switch.

This is the answer almost everyone gives — including, famously, a lot of mathematicians when the problem was first published.

It's wrong.