Monty's Dilemma

There is a TV game show in which the contestant is asked to choose one of three doors. Behind one of the doors is a whopping big prize, and behind the other two doors there is junk.

After the contestant chooses one of the doors, the game show host shows him what is behind one of the other two doors, always showing a "junker." Then the contestant is presented with the following dilemma:

Would you like to keep the door you chose,

or switch to the other ( still veiled) door?

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Monty’s Strategies

Let us pose a mathematical (probabilistic) problem from this dilemma.

Which of these three strategies is most likely to.lead the contestant to the winning door?

1) Just stay put, and keep the original door you chose, after the junk door is disclosed

2) Choose again by randomly selecting a door from the remaining two closed doors.

3) Choose again by switching from the door you chose to the other closed door.

How could we simulate the playing of Monty's game many times over?

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

The answer is (3) – You should always switch! The first door has a 1/3 chance and the second door has a 2/3 chance.

This problem first appeared in the Ask Marilyn column in Parade magazine. It has also been featured in the Mathematics Teacher and other math journals. Here is Marilyn vos Savant's reply to a reader who said that the odds would go up to 1/2: