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User talk:Martin Hogbin/Monty

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Car hidden behind Door 1 Car hidden behind Door 2 Car hidden behind Door 3
Player initially picks Door 1
Player has picked Door 1 and the car is behind it Player has picked Door 1 and the car is behind Door 2 Player has picked Door 1 and the car is behind Door 3
Host opens either goat door Host must open Door 3 Host must open Door 2
Host opens Door 2 half the time if the player picks Door 1 and the car is behind it Host opens Door 3 half the time if the player picks Door 1 and the car is behind it Host must open Door 3 if the player picks Door 1 and the car is behind Door 2 Host must open Door 2 if the player picks Door 1 and the car is behind Door 3
Switching loses with probability 1/6 Switching loses with probability 1/6 Switching wins with probability 1/3 Switching wins with probability 1/3
Switching loses with probability 1/3 Switching wins with probability 2/3

Solution

Say you initially pick either of the goats, the host must open the other door with a goat behind it in order to reveal a goat. So you have one door with a goat behind it and the host has opened the other door which also has a goat behind it. The remaining door therefore must have the car behind it. So if you swap, you get a car.

If you initially pick the car, it makes no difference which door the host opens, any other door has a goat behind it. so if you swap, you get a goat.


Player has picked a goat Player has picked a car
Player has picked Door 1 and the car is behind Door 2 Player has picked Door 1 and the car is behind Door 2
The host opens the other goat door The host opens the either goat door
Host must open Door 3 if the player picks Door 1 and the car is behind Door 2 Host must open Door 3 if the player picks Door 1 and the car is behind Door 2
Players who chose a goat and switch must get a car Players who chose a car and switch must get a goat
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