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# The Monty Hall problem

You are a contestant on a game show. At the show's conclusion you are presented with three doors, each of which conceals a prize. Behind one of the doors is a car. Behind each of the other two doors is a goat.

After you have selected one of the doors, the host will open one of the two remaining doors to reveal a goat. At this point you will have the option of opening the door you originally selected and taking the prize behind it, or switching to the remaining unopened door and going home with the prize it conceals.

Is it in your best interest to switch? Will it improve your odds?

The probability that you picked the correct door on your first guess is one in three.

The probability the host will open a door with a goat (other than the door you picked) is 100%, as defined in the problem. So there's a one in three chance the car is behind the door you chose, and a zero in three chance it is behind the door the host picked. Therefore, there must be a two in three chance it's behind the only other door.

So, yes, you should switch.

The host, in effect, is giving you a hint where the prize is. He's providing information about the state of the system which you did not have before. Every single time he must open a door you did not choose, and reveal a goat.