Many state governments use lotteries to raise money for public programs. In a common type of lottery, a customer buys a ticket with a three-digit number from 000 to 999. A machine (such as one with bouncing balls numbered 0 to 9) then selects a number in this range at random. Each ticket bought by a customer costs $1, whether the customer wins or loses. Customers with winning tickets are paid $500 for each winning ticket.
(a) Sketch the probability distribution of the random variable X that denotes the net amount won by a customer. (Notice that each customer pays $1 regardless of whether he or she wins or loses.)
(b) Is this a fair game? (See Exercise 31 for the definition of a fair game.) Does the state want a fair game?
(c) Interpret the expected value of X for a person who plays the lottery.