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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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Solution Explanation to Puzzle Answer Comment 42 of 42 |
(In reply to Puzzle Answer by K Sengupta)

Let the boxes be P, Q and R.

Then, we have the following cases:
(+) denotes that the individual gets the prize
(--) denotes that he individual does NOT get the prize

Prize is       Individual       If the individual      If the individual  
In Box         picks  Box       switches box         does not switch
----------        -----------------      ------------------------      ---------------------------
 P                    P                        --                                 +
 P                    Q                        +                                 --
 P                    R                        +                                 --
 Q                    P                        +                                 --
 Q                    Q                        --                                 +
 Q                    R                        +                                 --
 R                    P                        +                                 --
 R                    Q                        +                                 --
 R                    R                        --                                 +

From the chart given above, we observe that:
--> If the individual  switches, then the probability that he will WIN the prize is 6/9 = 2/3
--> If the individual does NOT switch, then the probability of his winning the prize is 3/9 = 1/3
Accordingly,  switching boxes would double this likelihood of winning th prize, and consequently,  he definitely should make the switch.

Edited on June 27, 2022, 11:58 pm
  Posted by K Sengupta on 2022-06-27 23:51:58

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