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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): NO! 50/50! All possibilites shown! | Comment 29 of 42 |
(In reply to re(2): NO! 50/50! All possibilites shown! by jduval)

I haven't exactly been following this conversation here, but I'm going to agree with Dustin.

Dustin listed these equally likely possibilities.

You pick box one, 1 has prize.  Host reveals box 2 empty.  stay
You pick box one, 1 has prize.  Host reveals box 3 empty.  stay
You pick box one, 2 has prize.  Host reveals box 3 empty.  switch
You pick box one, 2 has prize.  Host reveals box 3 empty.  switch
You pick box one, 3 has prize.  Host reveals box 2 empty.  switch
You pick box one, 3 has prize.  Host reveals box 2 empty.  switch


Yes, you are absolutely correct that some possibilities are listed twice.  That is because those particular possibilities are twice as likely.  If we eliminated the repeats as you suggest, we would come to some incorrect conclusions.  Here is the list without the repeats:

You pick box one, 1 has prize.  Host reveals box 2 empty.  stay
You pick box one, 1 has prize.  Host reveals box 3 empty.  stay
You pick box one, 2 has prize.  Host reveals box 3 empty.  switch
You pick box one, 3 has prize.  Host reveals box 2 empty.  switch

Based on this list, we can conclude that box 1 is more likely to contain the prize when we pick box 1.   Surely you'll agree with me that box 1 shouldn't be more likely to contain the prize just because we picked it?

Lists of equally likely possibilities can be misleading.  Just because there are two different possibilities doesn't mean each is equally likely.

  Posted by Tristan on 2005-03-02 23:31:06

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