The Monty Hall problem is a fascinating puzzle that challenges our intuition about probability. In this blog post, I will explain what the problem is, why the correct answer is surprising, and how to understand it using some simple logic and math.
The problem is based on a game show called Let’s Make a Deal, hosted by Monty Hall. The game goes like this: you are shown three doors, behind one of which is a car, and behind the other two are goats. You get to pick one door, but before you open it, Monty will open one of the other two doors and reveal a goat. He then gives you a chance to switch your choice to the remaining unopened door. Should you switch or stick with your original choice?
Most people would think that it doesn’t matter whether you switch or not, because after Monty reveals a goat, there are only two doors left, so the probability of the car being behind either door is 50%. However, this is wrong. The correct answer is that you should always switch, because that way you have a 2/3 chance of winning the car, while sticking with your original choice gives you only a 1/3 chance.
Why is this so? The key is to realize that Monty’s action of opening a door and showing a goat is not random. He knows where the car is, and he will never open that door. Therefore, he is giving you some information about where the car is not. This information changes the probability of the remaining doors.
One way to see this is to imagine that there are 100 doors instead of 3. You pick one door, and then Monty opens 98 other doors, all showing goats. He then asks you if you want to switch to the other unopened door. In this case, it should be clear that you should switch, because the probability that you picked the right door out of 100 is very low (1/100), while the probability that the other door has the car is very high (99/100). Monty has essentially narrowed down the possibilities from 100 to 1.
The Monty Hall problem shows us that our intuition about probability can be misleading sometimes. It also shows us how to use logic and math to analyze situations involving uncertainty and information.
