Paying to open boxes
Five closed boxes stand in a row. One holds 150 dollars and the others are empty; the prize is equally likely to be in any box. You pay a fee for each box you open, and you keep opening boxes until you find the money.
Which fee makes the game fair?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
How many boxes do you open on average?
The prize is equally likely to be found on the first, second, ..., fifth opening.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. You always win the 150 dollars in the end; the only question is how many fees you pay.
Step 1. Opening boxes in any order, the prize is found on opening or with equal probability . So you open boxes on average.
Step 2. The game is fair when the expected fees equal the prize:
Note. You never need to open the fifth box: if the first four are empty, the money is in the last one. If you may stop paying then, you open boxes on average and the fair fee rises to . The question as stated counts every box opened.