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 Pick a box! (Posted on 2002-03-28)
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: NO! 50/50! All possibilites shown! | Comment 24 of 42 |
(In reply to NO! 50/50! All possibilites shown! by jduval)

I'm sorry, you're wrong.

There are 9 equally likely possibilities:

You pick box one, 1 has prize.  Host reveals box 2 or box 3 empty.  Either way, 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

You pick box two, 1 has prize.  Host reveals box 3 empty.  switch
You pick box two, 2 has prize.  Host reveals box 1 or box 3 empty.  Either way, stay
You pick box two, 3 has prize.  Host reveals box 1 empty.  switch

You pick box three, 1 has prize.  Host reveals box 2 empty.  switch
You pick box three, 2 has prize.  Host reveals box 1 empty.  switch
You pick box three, 3 has prize.  Host reveals box 1 or box 2 empty.  Either way, stay

As you can see, in only three out of the nine cases, it is best to stay.  In six cases, you should switch.

In my example, cases 1, 5, and 9, it doesn't matter what the host does.  In your example you counted each of these as two seperate cases, and you could do that as long as you realize they no longer happen 1/9 of the time, they happen 1/18 of the time.  For example:

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

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

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

The probability is still 2/3 that switching will get you a prize.

Edited on March 1, 2005, 5:34 pm
 Posted by Dustin on 2005-03-01 17:30:30

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