The Goat Behind the Door

Nov 13, 2021
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

You should always switch your choice. The answer might seem a bit surprising. Why should you switch?

This problem is called the Monty Hall Problem. It's a lesson in expected value and randomness. First – the assumptions in the problem:

  1. The host will open a door that wasn't picked by the contestant.
  2. The host will open a door that contains a goat.

The next step to understanding the paradox is to just show the possible outcomes. Let's say you picked Door 1. There are three possibilities.

Scenario a: Car – Goat – Goat
The host can open either door 2 or door 3 (both goats). You guessed correctly, and if you stay with your choice, you win the car.

Scenario b: Goat – Car – Goat
The host opens up door 3, the only other door with a goat. If you switch, you win.

Scenario c: Goat – Goat – Car
The host opens up door 2, the only other door with a goat. If you switch, you win.

So you have a 2/3 chance of winning if you switch. Consider that your original guess had a 1/3 chance of being correct. You should always switch.

If you consider the case where the host opens up a random door that isn't yours, e.g., a door that could contain the car, your probability of winning is 1/2 whether or not you switch doors. So even in the absence of you knowing whether or not the host has chosen the door randomly, you have nothing to lose from switching.

Many of us biased towards sticking with the same door, even when we are presented with new information. This is called the endowment effect in behavioral economics.

The endowment effect is the tendency to retain an object you own rather than acquire the same object you do not own.