Chapter 6, Problem 53CP

Straight or boxed? Consider a Pick-3 lottery such as the one described in Exercise 6.7. Suppose your birthday is May 14, and like many people, you decide to bet $1 on your birthday number (i.e., 514) on your birthday. There are two options to play, straight or boxed. If you choose to play straight, you win $500 if and only if the number chosen is 514. If you choose to play boxed, you win $80 if the number chosen contains the digits 5, 1, and 4 in any order. Which option would you prefer to play and why?

6.7 Playing the lottery The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999. The lottery selects a three-digit number at random. With a bet of $1, you win $500 if your number is selected and nothing ($0) otherwise. (Many states have a very similar type of lottery.) (Source: Background information from www.ohiolottery.com.)

1. a. With a single $1 bet, what is the probability that you win $500?
2. b. Let X denote your winnings for a $1 bet, so x = $0 or x = $500. Construct the probability distribution for X.
3. c. Show that the mean of the distribution equals 0.50, corresponding to an expected return of 50 cents for the dollar paid to play. Interpret the mean.
4. d. In Ohio’s Pick 4 lottery, you pick one of the 10,000 four-digit numbers between 0000 and 9999 and (with a $1 bet) win $5000 if you get it correct. In terms of your expected winnings, with which game are you better off—playing Pick 4, or playing Pick 3? Justify your answer.