The Monty Hall Problem

The Monty Hall problem is a probability puzzle named after the host of the US game show Let's Make a Deal. It's a puzzle that has fooled some of the best mathematicians in the world, but it rarely fools pigeons. The puzzle is as follows:

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?

Give it a go below. On the off-chance you prefer goats to cars, I've replaced the car with a sack of cash, with which you can buy a car or multiple goats.

Results

Simulation

The simulation will start by picking a door at random and then alternate between sticking and switching for the final choice.

Start

Explanation

As you can see from the simulation results, you are twice as likely to win if you switch from your initial choice.

This can feel counterintuitive, because in isolation, the second choice is between two doors with two available outcomes and is therefore a 50-50 choice.

However it is important to remember that the host is not opening doors at random. He will never open the door you chose, and he will never open the door with the car/cash. He will always open a door with a goat and most of time, you will have selected one of the goats, so he will be forced to choose the one remaining goat.

Ultimately the probability that sticking will result in a win, is the probability that your initial choice was correct, which we know was 1/3. When the host eliminates all but one of the other two doors, he is gifting you the alternative 2/3 probability if you switch.

It is easier to see this if you increase the number of doors. Let's say you start with ten doors and after your initial choice Monty opens eight doors, revealing eight goats, leaving your door and one other. Do you think your initial 1 in 10 choice was correct, or would you rather switch? Try it out.