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Pick a box! (Posted on 2002-03-28) Difficulty: 3 of 5
You are shown three boxes, and told that one of them contains a prize. You are then asked to pick one box, and if that box is the one with the prize, you will win it. After picking a box, you are shown that one of the other two boxes is empty, and offered a chance to change your selection.

Should you do this? Would changing your choice to the other remaining box affect your odds of winning? Why or why not?

See The Solution Submitted by levik    
Rating: 4.2857 (14 votes)

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re(3): YES! ~33/66! All possibilites shown! | Comment 26 of 42 |
(In reply to re(2): YES! ~33/66! All possibilites shown! by Dustin)

I still disagree.  Firstly, I did not yell at you.  So there is no reason to yell at me and everyone who reads these posts.<o:p></o:p>

I agree that your odds of being correct the second time (changing your choice) is higher.  But, read on to understand why I disagree then with your conclusion (and Levik's).<o:p></o:p>

You failed to include the entire chart.  Look at the titles at the top of the two sides of the chart, Switch Yes and Switch No.  I took into account all of the possibilities.  Now even looking at your chart 2Y, 2N, 2Y*0.5=1Y&1N, 2N*0.5=1N&1Y.  That is still 4Y and 4N.  (Or even 2Y*0.5=1Y, 2N*0.5=1N, that is 3Y and 3N.)  That is still 50/50.  Either way you are still supporting my argument Thanks.<o:p></o:p>

Let's look at the two choices independently.  In the first choice you have a 1/3 chance of getting the prize.  No matter what box the host shows you or no matter what you pick, you still have a 1/3 chance of being right.<o:p></o:p>

Now for the second choice.  It is independent of the first choice.  Now there is one box with a prize and one box without.  If you switch your choice there is still one box with a prize and one box without a prize.  50/50.  If you do not change your choice, there is still one box ith a prize and one box without.  You still have a 50/50 chance of being right.  The two separate choices are independent of each other.<o:p></o:p>

Let's say the host does not show you anything.  And offers you the chance to change your choice.  There is still only one box with a prize and two boxes without the prize. Whether you change your first choice or not your odds are still 1/3.<o:p></o:p>

In the second choice your odds of being correct is indeed increased to 50/50, whereas in the first choice your odds were only 1/3.  But the HOST by showing you an incorrect box is increasing your odds, and not you by CHANGING YOUR CHOICE.  (Emphasis not yelling).<o:p></o:p>

Do you agree Dustin (everyone).  Or do you still feel I am out to lunch.<o:p></o:p>

  Posted by john on 2005-03-02 16:08:25
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