Mitan Ltd
Game-Show puzzle - explanation
Simple explanation
When you first pick one box, A, from the three, there is a 1-in-3 chance that A is the winning box, and a 2-in-3 chance that one of the other boxes B/C contain the prize.
Suppose you are now given the choice to stick with box A, or to pick both the other boxes B/C. If either box B or C contains the prize then you win. There is clearly a 2-in-3 chance that B/C contains the prize, so you should change your mind.
The game show host effectively gives you this choice byopening B or C and shows that it is empty. The host knows where the prize is, and can be sure of opening an empty box.
The Answer
You should change your mind and switch from your original choice of box A to the remaining box B. This will increase your chance of winning from one-in-three to two-in-three.
Many people find the answer hard to believe, which is more interesting than the answer itself. Calculating the odds is not complicated, but there are other plausible but incorrect lines of reasoning..
What Most People Think And Do
Many people stick with their original choice, believing that since the prize is equally likely to be in any of the boxes, it makes no difference to stick with box A or change to box B, and that being allowed to change your selection is just showmanship to add excitement to the game .
Psychology is another factor. Choosing a box is emotional, since getting it wrong means losing a valuable prize. Changing your mind may look weak and indecisive., which might be worse than just choosing the wrong box.