Today is the day. You’ve been waiting for this moment ever since you got accepted to join. Standing on stage, you are participating in your favorite TV show: “Let’s Make a Deal,” hosted by none other than the beloved Monty Hall. You glance over at the three huge, colorfully decorated doors, each with a number on it, one through three.
Hall introduces you to the crowd, and then explains how this game works. Behind two doors are goats, but the other one has a brand new car. However, these doors are random, and you don’t know which is which. You must choose one of the doors and try to get that prize.
Hall gives you the choice, and you pick door three. He asks you if you’re sure that you want what’s behind that door. Before you can respond, the second door opens, and you see a goat inside. You think about your options. He asks you if you want to keep your door, number three, or switch to door one. After thinking about it, even with all the pressure on stage, you decide to switch to door one. The door rolls up as you see flashes of light and a brand-new red Porsche 930. The audience applauds as Hall congratulates you on your new car!
But there’s a trick that can increase your chances of winning the car above a 50 percent chance. And that trick is to always switch to the other door, every time. This seems wrong! There are two doors, one with a goat and one with a car, so it should be 50-50. But no. If you switch every time, your odds of winning go up to two-thirds! But why? Why are your odds two-thirds and not just a half? To see why, let’s go back to the beginning. When you first pick, you have a one-third chance of the car being in that door.
This means that the other two doors have a combined two thirds chance of having the car. But when you are shown that one of the other two doors has a goat behind it, the odds still remain the same. One third for the one you chose, and two thirds for the other door you didn’t, thus making it beneficial to switch every time as it increases your chances of winning the car. If that doesn’t make much sense, look at it this way. Again, suppose you pick door one this time. Keep in mind that you’ve just been shown a door that has a goat behind it, so you only can switch to one other door.
If the car is behind door 3 and you switch, you win. If it’s behind door 2 and you switch, you win. If it’s behind door 1 and you switch, you lose. This applies to all series of doors, every time. As you can now see, probability doesn’t always come naturally to the human mind correctly, and it can be deceiving at times.