Expectations
A game costs $150 to play. In this game, you roll a fair six-sided die repeatedly until each of all the six numbers has been rolled at least once. You are then paid 10 times the number of rolls you made.
For example, if the rolls were 3, 5, 4, 3, 2, 5, 1, 4, 1, 3, 6, then you would get dollars.
Including the price to play, what is your expected value in this game?
The answer is -3.
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1 solution
The expectation for the number of rolls it will take for all six numbers to turn up is as follows:
One of the numbers is certain to turn up on the first roll. There are then 5 numbers out of 6 that will add a new number to the list, so you add (6/5) rolls. Similarly, you add (6/4), (6/3), 6/2), and (6/1).
Total expected number of rolls = 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.7
At $10 per roll, your expected winnings = 14.7 * 10 = 147
Since you pay $150 and expect back $147, your mathematical expectation is -3.