# Lucky Lottery Agency

Lucky Lottery Agency

Our state lottery agency, which we will call the Lucky Lottery Agency, is new and only offers three games. We hope to offer more games in the future, but we must keep the Agency running and in good finances for now.

Daily 3 The Daily 3 game is a daily game, drawn every day except for Saturday and Sunday. It consists of three sets of balls, each numbered from 0 through 8 (9 is omitted due to its visual similarity to  6). One ball is drawn from each set giving a 3-digit winning number. All numbers must be in the correct order to win, called a “straight” draw by most states’s lottery agencies. A ticket costs \$1. The prize for getting all three numbers correct is \$500.

Daily 4 The Daily 4 game is a daily game, drawn every day except for Saturday and Sunday. It consists of four sets of balls, each numbered from 0 through 9 (the Agency learned to differentiate 6 from 9 using lines when these machines were built). All numbers must be in the correct order to win, called a “straight” draw by most states’s lottery agencies. A ticket costs \$1. The prize for getting three numbers correct is \$100 and the prize for getting four correct is \$5000.

Lucky Ball Lucky Ball is a big game. It consists of one pool of numbers from 1 to 47 and a second pool from 1 to 37. Five balls are drawn from the first pool without replacement. A sixth ball, called the Lucky Ball, is drawn from the second pool. As the numbers are drawn without replacement, order is irrelevant. A ticket costs \$1. The prizes amounts are given in table 1. The jackpot starts at \$10,000,000. It grows each week until there is a winner, then returns to the starting value.

Problems
Each question is worth four points.
1. How many different combinations are possible with the Daily 3 game? The Daily 4 game?
2. Assuming each possible outcome is equally likely to appear, how much money, on average, will the Agency receive from each Daily 3 ticket, after paying winners? Daily 4 ticket? lucky lottery agency 2 Matches Lucky Ball Prize
5 Yes Jackpot
5 No \$100,000.00
4 Yes \$10,000.00
4 No \$1,000.00
3 Yes \$100.00
3 No \$10.00
2 Yes \$10.00
2 No \$5.00
1 Yes \$5.00
1 No \$1.00
0 - None
Table 1: Payout table for Lucky Ball
3. Due to a quirk in the Agency’s charter, the Agency must pay for all it’s operating expenses (except prizes) from the Daily 3 and Daily 4 games. This year, the Agency expects to spend \$4 million dollars in operating expenses. Create a linear inequality showing how many tickets from the Daily 3 and Daily 4 must be sold to met the \$4 million minimum, using the results of te last question.
4. You own a small store with a lottery terminal. As part of a profit sharing program with the Agency, you receive \$3 for every winning Daily 3 ticket you sell and \$20 for every winning Daily 4 ticket you sell. Last month, you sold 13 winning tickets and received \$107 as part of the revenue sharing program. How many winning Daily 3 and Daily 4 tickets did you sell?
5. The Agency has decided to add a “box” option for winning the Daily 3, where the numbers can be matched in any order. If the Agency has selected a box prize of \$10, what is the new average
the Agency will receive from each Daily 3 ticket after paying off winners?
6. In the Lucky Ball game, what the probability of winning the jackpot?
7. What are the combined probability of winning any prize in the Lucky Ball game?
8. In the Lucky Ball game, when a jackpot is won, the amount is paid in equal installments over 30 years to the winner through an annuity, unless the winner accepts the “cash option,” where the
winner receives the amount that would be put into the annuity. If the jackpot is \$30,000,000 and the annuity’s interest rate is 2%, compounded annually, what is the amount of the cash option? lucky lottery agency 3
9. What is the expected value, that is, how much money do you expect to win, of a Lucky Ball ticket before the drawing, assuming a jackpot of \$10,000,000, assuming there is only one jackpot winner?
10. At what size jackpot does the expected value become positive, assuming there is only one jackpot winner?