Monty Hall paradox

The Monty Hall Paradox is a problem concerning probability, which reveals counterintuitive results.

The Problem:

You are in a game show. There are three curtains. Behind one of the curtains is a car, and behind the other two are goats. The game show host knows which curtain the car is behind. You are asked to pick a curtain, and will be given the prize behind it. Right after you pick, however, the host reveals one of the curtains you did not pick, which has a goat behind it (because he knows which curtain the car is behind, he won't accidentally reveal the car). The host then asks you whether you would like to consider switching to the remaining unopened curtain. The question is: Do you stay, do you switch, or does it even matter?

The Answer:

Yes. Even though it looks like a 50% chance of getting the car, you will actually find that you can double your probability just by switching to the other curtain. To understand why, we have to take a look at what happened at the start. You picked one of the curtains, knowing that there was a 1/3 chance of picking the car and a 2/3 chance of picking a goat. Notice that when the host reveals one of the goats, it does not affect your choice. There is still the exact same uncertainty of the item behind the curtain. Because the host knows where the goat is, he will always reveal a goat. If you stick with your choice, it is basically the same thing as picking 1 curtain out of 3 and hoping to get the car which has a 1/3 chance. However, if you switch, then there is a 2/3 chance that you switched from the goat to the car.

What if there were more curtains?

Let's say that there are 100 curtains. You pick one, and then the host reveals 98 of the remaining 99 to be goats. Do you stay, or switch? Of course, you switch since the probability that you picked the car at first is ridiculously small, so you can almost guarantee that the curtain you currently have does not have the car behind it, and therefore the remaining curtain must almost always have the car behind it.

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