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 or Door 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 the door 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!
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.
In 1990, a similar question appeared in a letter to Marilyn vos Savant's Ask Marilyn column in Parade (which comes in some Sunday newspapers). Marilyn gave the correct answer, but many readers (including mathematics professors) believed that this was incorrect. So don't feel too bad if you got it wrong when you answered it for yourself. But now you know!
Source :
http://www.me.ucsb.edu/~moehlis/21.html
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Bhushan S.
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