7. There are some tokens in a bag. Each token has a number ( a positive integer) printed on it. Ernesto does not know how many tokens are in the bag, and the only thing he knows about the numbers on them is that the mean of the numbers is 3 and the variance is 2.
Ernesto is going to play a game in which he draws 1 token from the bag and is paid an amount in dollars equal to the square of the number printed on the token that he draws. For instance, if he draws a 2 he will receive $4, but if he draws a 5 he will receive $25.
What is the maximum that Ernesto should be willing to pay to play this game if he wants to be sure that on average, if he plays very many times, he shouldn't lose money?
This shows how to calculate how much should be paid to play a given game involving probabilities.