# Probabilities in the Case of Power-Ball

The game of PowerBall goes like this: 5 white balls are drawn from one drum that contains 42 balls. One red ball (the powerball) is drawn from a separate drum that contains 20 balls. To win the jackpot, you must have all 5 white balls correct as well as the red powerball. No balls are replaced after being drawn. The order you have your numbers in does not matter. For example, if the numbers drawn from the white drum were 9, 8, 1, 3, and 6 and your numbers were 1 3 6 8 9 , you would still be a winner. Express all final answers in either lowest fraction form, or a decimal in standard form.

a. Find the probability of being a jackpot winner. You must show all work to show how you and your partner got the answer. There is more than one way to do this problem.

b. Would it make a huge difference in the probability if the order of the numbers did matter? (meaning you had to pick the 5 numbers in the same order as they were drawn from the drum) Find this probability of winning the jackpot if the order did matter and compare it to part a. You must show all work to show how you got your answer.

c. You can win $500,000 if you get all the white balls correct(in any order), but get the red ball wrong. What are the chances of winning $500,000? You must show all work to show how you got your answer.

d. Do you have a better chance of winning the $500,000 as explained above, or playing a lottery that doesn't even have the red powerball at all. Assume everything else about this other lottery is the same as explained in the beginning of this problem. You must show all work to show how you got your answer.

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#### Solution Preview

a. Since we will pick 5 white balls from 42 balls, there are C(42,5) ways to do so where C is the notation of combination. The reason why we use combination is that the order does not matter. Similarly, since we will pick 1 red ball from 20 balls, there are C(20,1) ways to do so. Since there is only 1 jackpot number, the probability of being a jackpot winner is

1/[C(42,5)*C(20,1)]=1/1701336

b. Yes, it would make a huge difference ...

#### Solution Summary

The solution gives detailed steps on calculating probabilities in the case of power-ball under different conditions.

Powerball Winning Combinations, Prizes, and Probabilities

Case Study: The Powerball

1. Review the case study on page 157 of the textbook. See other attachments

2. Recall that for a single ticket a player first selects five numbers from the numbers 1-55 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 the 55 white balls numbered 1-55, and one red Powerball is drawn randomly from 42 red balls numbered 1-42. To win the jackpot, a ticket must match all the balls drawn. Prizes are also given for matching some but not all the balls drawn.

3. Click Table of Powerball Winning Combinations, Prizes, and Probabilities (or see p. 231 in the textbook) for data about Powerball and probabilities of winning. Then answer the following questions.

a. If you purchase one ticket, what is the probability that you win a prize?

b. If you purchase one ticket, what is the probability that you don't win a prize?

c. If you win a prize, what is the probability it is the $3 prize for having only the Powerball number?

d. 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?

e. If you were to buy one ticket per week, approximately how long should you expect to wait before winning a prize?