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Consider the following variation of the final round of the classic TV game show

Let's Make A Deal:

There are three doors, and behind one of them is a car, while behind the other

two are goats. If you choose the door with the car behind it, you win the car.

Now, say you choose Door 1. The host Monty Hall then opens either Door 2 orDoor 3, behind which is a goat. (He knows what is behind each door, and never

opens the door with the car behind it.) Monty now gives you the choice: do you

want to stick with Door 1, or switch to the other door. What should you do? Or

does it matter?

A similar question is posed to Ben Campbell (played by Jim Sturgess) by

Professor Micky Rosa (played by Kevin Spacey) in the movie "21". Without

hesitation Ben answers this correctly, which convinces Professor Rosa that Ben

would be a good addition to their "card counting team". Before reading on, try

to answer this yourself.

One solves this problem by comparing the probability of choosing the car if you

stick with your original choice to the probability of choosing the car if you switch

after Monty opens the one door. Note that the car has an equal probability of 1/3

of being behind Door 1, Door 2, or Door 3.

First, suppose that your strategy is to stick with your original choice of Door 1.

Then you only win if the car is behind Door 1, so that your probability of

winning is 1/3.

Next, suppose that your strategy is to switch doors. We break this into three

cases:

If the car is behind Door 1, Monty will open either Door 2 or Door 3 to reveal a

goat. You switch to the other of Door 2 or Door 3, and in either case you switched

to a door with a goat behind it (remember, the car is behind Door 1).

If the car is behind Door 2, Monty will open Door 3. This is because he always

opens a door with a goat behind it, and he can't open Door 1 because that was

your original choice. So the only door you can switch to is Door 2, which is thedoor with the car behind it. Ding! You win!

If the car is behind Door 3, Monty will open Door 2. This is because he always

opens a door with a goat behind it, and he can't open Door 1 because that was

your original choice. So the only door you can switch to is Door 3, which again is

the door with the car behind it. Ding! You win!

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So if your strategy is to switch doors, you win 2/3 = 1/3 + 1/3 of the time.

(Remember, the probability is 1/3 that the car is behind any particular door.)

Therefore, a better strategy is to switch doors - the calculated probabilities

indicate that you are twice as likely to win if you do this! Ben's correct answer in

the movie "21" indicates that he is a good person for "counting cards". Not only

does it show that he is clever, but it also demonstrates that he realizes that it is

best to go with the choice which maximizes your probability of winning. This

realization is essential to the success of "counting cards" for Blackjack.