Suppose that you are playing a game in which a Genie lets yo

Suppose that you are playing a game in which a Genie lets you choose one of three doors. Behind one door is a million dollars, behind the other two doors there are goats. After you make your choice, the Genie does not open the door right away. He first opens one of the two doors that you did not select and reveals a goat. He gives you the choice to switch your choice, or to keep your choice unchanged. What do you choose and why? Derive with proof the conditional probabilities of each choice.

Solution

Yes I choose to switch.

Proof:

Let\'s assume the Prize is behind door 1.

Following are what would happen if the player had a strategy of NOT switching:

>> Picks Door 1 -> Win
>> Picks Door 2 -> Loss
>> Picks Door 3 -> Loss

Following are what would happen if the player had a strategy of switching:

>> Picks Door 1 -> Host reveals goat behind door 2 or 3 -> Player switches to other door--> Loss
>> Picks Door 2 -> Host reveals goat behind door 3 - > Player switches to door 1 --> Win
>> Picks Door 3 -> Host reveals goat behind door 2 --> Player switches to door 1 --> Win

So, by not switching the player has 1/3 chance of winning.
By switching , player has a 2/3 chance of winning.

Therefore,
Player should definately switch.