The Monty Hall Problem

This puzzle is called the Monty Hall Problem.

You should switch because if you do, you have a 2 in 3 chance of winning. If you don't switch, the probability of winning is 1 in 3.

The intuitive answer for many people is that the probability will be 50/50 after the host opens one of the doors. But our intuition misleads us here, as it does in many cases.

You can try out Monty Hall problem simulators online to see for yourself. If you consistently don't switch doors after enough rounds, you will see that you win one-third of the time, and if you do, it will be two-thirds.

Even though these simulators can be used to prove that the answer is correct, it still doesn't feel right. How can we make it make sense?

Let's change it to nine doors.

Behind one door is a million dollars, and behind 8 of them, toilet paper. Your odds aren't great anymore, but you choose a door with a 1 in 9 chance of winning the million dollars. Then the host, who knows what's behind each door, opens seven other doors to reveal toilet paper. He asks if you'd like to switch.

Now it does feel intuitively right to switch. The odds of winning were only 1 in 9 when you first chose. They are 8 in 9 if you switch.

The Monty Hall problem is a great example of how our intuition can lead us astray. It was published in a magazine in 1990, and the answer just felt so wrong that 10,000 readers wrote to the magazine saying they had made a mistake!