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.)
a. With a single $1 bet, what is the probability that you win $500?
b. Let X denote your winnings for a $1 bet, so x = +0 or x = +500. Construct the probability distribution for X.
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.
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 in which you win $500 for a correct choice of a three-digit number? Justify your answer.