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Statistics : Probability of Winning a Powerball Lottery

The Powerball

For a single ticket, a player first selects five numbers from the numbers 1-53 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 53 white balls numbered 1-53, 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. Table 4.18 displays the number of matches, the prizes given, and the probabibilities of winning.

4.18
Matches Prize Probability
5 + 1 Jackpot 0.00000001
5 100,000 0.00000034
4 + 1 5,000 0.00000199
4 100 0.00008164
3 + 1 100 0.00009359
3 7 0.00383716
2 + 1 7 0.00143503
1 + 1 4 0.00807207
0 + 1 3 0.01420684

Some things to note about table 4.18:
-In the 1st column, an entry of the form w + 1 indicates w matches out of the five plus the Powerball;
one of the form w indicates w matches out of the five and no Powerball.
-The prize amount for the jackpot depends on how recently it has been won, how many people win it,
and the choice of Jackpot payment.
-Each probability in the 3rd column is given to eight decimal places.

Let E be an event having probability p. In independent repetitions of the experiment, it takes, on average, 1/p times
until event E occurs. We apply this fact to the Powerball. Suppose that you were to purchase one Powerball ticket
per week. How long should you expect to win the jackpot? The 3rd column of Table 4.18 shows that, for the jackpot,
p= 0.00000001, and therefore we have 1/0.00000001= 100,000,000. So if you purchased one Powerball ticket per week,
you should expect to wait approximately 100 million weeks, or roughly 1.9 million years, before winning the jackpot.

Questions:

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

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Solution Summary

The Probability of Winning a Powerball Lottery is investigated. The solution is detailed and well presented.

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