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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?

 Submitted by levik Rating: 4.2857 (14 votes) Solution: (Hide) Switching your choice would actually double your odds, and would be the smart thing to do. Here's why: Let's say you start off by picking an arbitrary box. Let's call it box (A). Let's also call the other two boxes (B) and (C). Since the chance of the prize being in a particular box is 1/3, your selected box, (A), will be the winner one out of three times. Which means you will lose the other two times, when the prize will be in either box (B) or box (C). Now the problem states that you are shown that one of these boxes is empty. But of course one of them has to be empty since the prize cannot be in both of them. And of course the odds of the prize being in either (B) or (C) are not affected by revealing which box the prize is not in. So let's say the box revealed to be empty is (B). If the chance of the prize being in either (B) or (C) is 2/3, and we know that it's definitely not in (B), that means that we can say that the probability is 2/3 that it is in (C) - and 1/3 that it is in (A). Thus changing your choice raises your chance of winning from one out of three to two out of three. You can see the statistics of this here. Update: Here's a more intuitive explanation... At the time you make your initial guess and pick one box, the prize is either contained in that box, or not contained in it. If the prize is in the box you picked, changing your choice is obviously the wrong thing to do. However, if you were initially wrong, then the prize is in one of the other two boxes. After one of those boxes is revealed to be empty, the prize will most definitely be in the last remaining box, so you should definitely switch So you stay with your choice if you are initially correct, and switch if you were wrong. Now the chance that you were right with your first guess is 1/3, and that you were wrong - 2/3, so in two out of three cases a switch makes sense.

 Subject Author Date Explanation to Puzzle Answer K Sengupta 2022-06-27 23:51:58 Puzzle Answer K Sengupta 2022-06-27 00:05:02 Alternate explanation of the solution tomarken 2009-03-25 11:53:52 Two Solutions owl 2005-08-17 14:36:31 Solution Stephen Ticsay 2005-04-16 04:08:18 re: Cool analogy yocko 2005-03-08 07:38:34 6/9 Bractals 2005-03-08 07:12:10 Agree with solution? Problem simplified? john 2005-03-07 17:11:43 re: Cool analogy john 2005-03-07 16:52:33 Cool analogy Tristan 2005-03-04 03:01:28 re: Flat out wrong logic in solution. Tristan 2005-03-03 22:20:52 Flat out wrong logic in solution. john 2005-03-03 18:15:56 re(4): YES! ~33/66! All possibilites shown! Dustin 2005-03-03 00:15:47 re(3): NO! 50/50! All possibilites shown! Tristan 2005-03-02 23:31:06 re(2): NO! 50/50! All possibilites shown! john 2005-03-02 21:03:49 Disagree with solution! john 2005-03-02 16:18:09 re(3): YES! ~33/66! All possibilites shown! john 2005-03-02 16:08:25 re(2): YES! ~33/66! All possibilites shown! Dustin 2005-03-01 17:42:05 re: NO! 50/50! All possibilites shown! Dustin 2005-03-01 17:30:30 re: NO! 50/50! All possibilites shown! john 2005-03-01 17:24:54 NO! 50/50! All possibilites shown! john 2005-03-01 17:09:37 re: kinda easy john 2005-03-01 16:21:47 re: owl 2004-12-11 05:46:49 No Subject Iain 2004-05-09 14:24:09 No Subject logischer Verstand 2004-05-02 20:53:08 maybe Billy Bob 2004-03-28 10:29:41 kinda easy omar 2004-02-21 15:31:40 No Subject Cory Taylor 2002-12-17 10:17:09 re: hmmm levik 2002-11-06 10:14:07 hmmm danny 2002-11-05 11:35:50 Come on guys! Arcus 2002-10-29 09:25:40 re: 50% vs 67% levik 2002-10-17 09:01:31 50% vs 67% Jim Lyon 2002-10-16 13:08:30 re: Why 50% is not the answer. levik 2002-08-15 10:16:15 Hey Dulanjana 2002-08-14 20:55:02 Why 50% is not the answer. Paul 2002-08-14 11:14:20 50% is the ONLY answer. Mike 2002-05-10 15:32:06 Math vs. Puzzles Chris 2002-05-07 04:50:40 More info John 2002-05-01 06:46:13 More info to Re: No John 2002-05-01 06:44:48 Re: No John 2002-05-01 06:43:30 No D Willson 2002-04-23 22:19:13

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