Most people can readily understand that since you selected one of the three doors, if you stay, the probability of winning is 1/3. What happens if you switch? Assuming that the host always opens a door that does not have the car, and this is a crucial assumption, you have a 2/3 chance of winning if you switch.There are three equally likely possible orderings of the prizes behind the three doors, shown as A, B, or C.
Original Door Selected
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Door 1
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Door 2
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Door 3
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Actual Situation
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Order A
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Car
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Goat
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Goat
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Order B
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Goat
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Car
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Goat
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Order C
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Goat
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Goat
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Car
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Suppose the player has selected Door 1. If the car is behind Door 1, as in order A, the host will open either Door 2 or Door 3, and if the participant switches, he or she will get a goat. If the car is not behind Door 1, as in order B or C, then the host will open the remaining door that has a goat, and if the player switches, he or she will get the car. Only order A would result in a loss, that is, a goat. This same analogy also works if you start with the player selecting Door 2 or Door 3. The probability that player wins a car with a switch is . = 2/3