Question

The Powerball is a lottery sold in Arizona, Connecticut, Delaware, District of Columbia, Idaho, Indiana, Iowa, Kansas, Kentucky, Louisiana, Minnesota, Missouri, Montana, Nebraska, New Hampshire, New Mexico, Oregon, Rhode Island, South Dakota, West Virginia, and Wisconsin. It was introduced in April of 1992 and took its present form in November of 1997.

     Although the Powerball is a multi-state lottery game, it is not the first. That distinction goes to Lotto*America, which was created in 1988 when Iowa and six other states joined forces to offer a game with a large jackpot. Since the more people that play, the bigger the jackpots tend to be, multi-state lotteries offer larger prizes than those standing alone.

     A Powerball jackpot starts at $10 million and grows if no one wins it. Since, as we will see, the chance of winning a jackpot is small, the jackpot often grows to huge amounts, sometimes around $300 million. In case there are multiple winners, the jackpot is divided equally among them. Drawings take place on Wednesday and Saturday evenings at 9:59 P.M. and can be watched during the 10:00 P.M. news.

     Here is how the Powerball is played: A player first select five numbers from the numbers 1-49 and then chooses a Powerball number, which can be any number between 1 and 42. A ticket costs $1. In the drawing, five white balls are drawn randomly from 49 white balls numbered 1-49; and one red Powerball is drawn randomly from 42 red balls numbered 1-42.

     To win the jackpot, a ticket must match all of the balls drawn; there are other smaller prizes for matching some but not all of the balls drawn. What are the chances of winning the jackpot? What are the chances of winning any prize whatsoever? The following table provides the number of matches, the prizes given, and the probabilities of winning.

Matches	Prize	Probability
5+1	Jackpot	0.00000001
5	$100,000	0.00000051
4+1	$5,000	0.00000275
4	$100	0.00011262
3+1	$100	0.00011812
3	$7	0.00484285
2+1	$7	0.00165366
1+1	$4	0.00847500
0+1	$3	0.01355999

Here are some things to note about the above table.

● In the first column, an entry of the form w+1 indicates w matches out of five plus the Powerball; one of the form w indicates w matches out of five and no Powerball

● The prize amount for the jackpot depends on how recently it has been won, how many people win it, and choice of jackpot payment.  

● Each probability in the third column is given to eight decimal places.

Let E be an event having probability P. It can be shown that in independent repetitions of the experiment, it takes on the average times until event E occurs. We will apply this fact to the Powerball.

Suppose we were to purchase one Powerball ticket per week. Let’s find how long we should expect to wait before winning the jackpot. The third column of the above table shows that for the jackpot, P=0.00000001 and, therefore, we have 1/ 0.00000001 = 100,000,000. So, if we purchased one Powerball ticket per week, then we should expect to wait approximately 100 million weeks or roughly 1.92 million years, before winning the jackpot.

1. If you purchase one ticket, what is the probability that you win a prize? (1 mark)
2. If you purchase one ticket, what is the probability that you don’t win a prize? (1 mark)
3. If you win a prize, what is the probability it is the $3 prize for having only the Powerball number?
4. If you were to buy one ticket per week, approximately how long should you expect to wait before getting a ticket with exactly three winning numbers and no Powerball?
5. If you were to buy one ticket per week, approximately how long should you expect to wait before winning a prize? (1 mark)

Answer 1

a.

probability that you win a prize = 0.00000001 + 0.00000051+ 0.00000275 + 0.00011262 + 0.00011812 + 0.00484285 + 0.00165366 + 0.00847500 + 0.01355999

= 0.02876551

b.

probability that you don’t win a prize = 1 - probability that you win a prize = 1 - 0.02876551

= 0.9712345

c.

If you win a prize, the probability that it is the $3 prize for having only the Powerball number

= P(win $3 prize for having only the Powerball number | win any prize)

= P(win $3 prize for having only the Powerball number and win any prize) / P(win any prize)

= P(win $3 prize for having only the Powerball number) / P(win any prize)

= 0.00011812 / 0.02876551

= 0.004106306

d.

Probability of getting a ticket with exactly three winning numbers and no Powerball = 0.00484285

Let X be the number of tickets bought to win a three winning numbers and no Powerball.

X ~ Geometric(p = 0.00484285)

Number of weeks to wait a three winning numbers and no Powerball = 1/p = 1/0.00484285 = 206.49 weeks

e.

Let Y be the number of tickets bought to win a prize

Y ~ Geometric(p = 0.02876551)

Number of weeks to wait to win a prize  = 1/p = 1/0.02876551 = 34.76385 weeks