Monty Hall Problem

Consider a nightly gameshow with the following setup. There are three doors. Behind each of two of the doors is a goat. Behind the third remaining door is a car. On any given night, the car is `equally likely’ to be behind any one of the three doors. There are a host, who knows the location of the car, and there is a contestant, who does not know the location of the car. The contestant chooses a door. The host then opens one of the doors, behind which is a goat, and offers the contestant the chance to swap his or her door for the remaining door.

Should the contestant keep his or her door or swap it for the other? Here is my mathematical understanding of the problem and the corresponding solution.

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