You want to always switch the box. When you first pick the box, you had a 2 out 3 chance of losing (1 prize and 2 bad prizes). Your odds of winning are 1 in 3 or 33.33%. If you don't change your answer, this number can't change since the order of what is in the boxes does not change over time. However, If you switch boxes, your chance of losing is different. You now know which which box has one of the bad prizes, since it was shown to you, so you do not pick it (not allowed actually). There can only be one unknown bad prize remaining. The situation can now look like:
A. 1) bad, 2) bad 3) prize
B. 1) bad, 2) prize, 3) bad
C. 1) prize, 2) bad 3) bad
There are no other combinations.
You can go over each situation. Lets say you always pick the first column.
For A
Picked bad, 2 shown to you. You switch and win.
For B
Picked bad, 3 shown to you. You switch and win.
For C
Picked prize. 2 or 3 shown to you. You switch and lose.
So you win 2/3 or 66.65% of the time. You can repeat this with each door as your selection and the match is the same.
If you don't switch the odds are only 33.33%, but if you switch it jumps to 66.67%. It defies common sense, but the math works out. When you are given more info, like which box has one of the bad prizes, it changes the odds..
