We have created a Mathematica notebook which analyzes the strategic advantage of changing your choice of door (as opposed to staying with your original choice) when you play the game Let's Make a Deal. 

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Suppose you choose door Number 1.  After that, the gameshow host shows you what's behind one of the other doors you didn't pick, say, door Number 2.  He gives you the opportunity to change your mind, and choose door Number 3 instead of door Number 1.  Common sense tells us that there is nothing to be gained by switching to door Number 3.  After all, once one of the doors is opened, the probability that the prize is behind the door you originally chose is now ½ and the same is true for door Number 3.  Isn't that right?

 

However, if you run the simulation contained in the Mathematica notebook provided, you will see that common sense is wrong!  Switching to the other door will almost always improve your chances of winning.

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To obtain the Mathematica notebook which illustrates this conundrum, click on the Let's Make a Deal.zip icon directly above.  When prompted, download the zip file to your computer.

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If you are running Windows. we recommend that you download the zip file to your Downloads directory.  Before you extract the contents of the zip file, we suggest that you first create a directory on your PC into which your unzip utility will extract the zip file's contents.  Then proceed to unzip the file, telling your unzip program to deposit the extracted file in the directory you created.

 

If you are running on a Mac, depending on your settings when you download the zip file, it may be automatically unzipped.  If not, you will be prompted to open (unzip) the file.  When you do so, you will see that your Downloads folder now contains a file named lets lets Make a Deal.nb, which you can move to some other folder on your Mac.

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The file extracted from the zip file is a Mathematica notebook named lets Make a Deal.nb.