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
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.
Exercise 31
A game involving chance is said to be a fair game if the expected amount won or lost is zero. Consider the following arcade game. A player pays $1 and chooses a number from 1 to 10. A spinning wheel then randomly selects a number from 1 to 10. If the numbers match, the player wins $5. Otherwise the player loses the $1 entry fee.
(a) Define a random variable
that is the amount won by the player and draw its probability distribution. (Capital letters other than
can be used for random variables.) Use negative values for losses, and positive values for winnings.
(b) Find the mean of
. Is this a fair game?