You are a contestant on a game show hosted by Monty Hall in

You are a contestant on a game show (hosted by Monty Hall), in which you are presented with 12 doors. Behind 11 of these doors, there is a goat (i.e. a ‘bad prize’), but behind exactly 1 of these doors is $1 million.

The rules of the game are as follows. You will select one of the 12 doors. The host will then remove 9 of the other 11 doors (all of the doors he eliminates are guaranteed to have the goat behind them). You will then be given the chance to ‘stay’ with the door that you have already picked, or to ‘switch’ to one of the other two remaining doors. You will get whichever prize is behind the door you end up with.

a) Should you ‘stay’ or ‘switch’?

b) What is the probability of the $1 million being behind each of the three doors remaining in the second part of the game (the door you originally chose, plus the other two)? State your reasoning.

Solution

a) probability of the door choosen at 1 st round to have 1 million behind it is ( 1 / 12)......

at 2 nd stage, probability of any of the other 2 door remaining to have the 1 million behind it is 1/3.....

1/3 > ( 1/11)..so, it will be wise to switch!

b) probability of the $1 million being behind each of the three doors remaining in the second part of the game (the door you originally chose, plus the other two) = 0....

As the game clearly states that behind \" exactly \" 1 of these 12 doors is $1 million...