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Tuesday, July 29, 2008

A Quiz...

You are a participant in a quiz. The quizmaster shows you three closed doors. He tells you that behind one of these doors there is a prize, and behind the other two doors there's nothing. You select one of the doors, but before you open it the quizmaster deliberately picks out a remaining empty door and shows that there is nothing behind it. The quizmaster offers you a chance to switch doors with the remaining closed door.

The Question: Should you stick to your choice?

Answer: In this puzzle, you should not use your intuition, but let your common sense do the job: the chance that your first choice for a door was correct is 1/3; therefore, the chance that your first choice was wrong is 2/3.

So the chance that one of the remaining doors is correct is also 2/3. With the help of the quizmaster (who knows which door hides the price, and thus is able to open one of the remaining doors which does not contain the the price), you get to know which one of the remaining doors is incorrect. Now you also know which one of the remaining doors could be correct with a chance of 2/3!

Conclusion: You should switch doors, which doubles your chances!...

For the disbelieving few: consider the situation where there are 1000 doors instead of 3. After you have chosen one door, the quizmaster points out 998 of the 999 doors that are left, that do not contain the prize. Should you switch to the other remaining door? Of course! If, out of 999 doors, the quizmaster (deliberately) leaves that door, chances are very large (999/1000) that it is the right one!

For the still disbelieving few: write a computer program which simulates this quiz thousands of times, and you will see that the chances are double if you switch doors!...

If after all this, you still don't believe it (and trust me, you are not alone), please have a look at the first puzzle (on page 49) and its solution (on page 55) of Martin Gardner's article "A Quarter-Century of Recreational Mathematics" in the Scientific American of August 1998.