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Powerball Lottery. The Powerball lottery is played twice each week in 44 states, the District of Columbia, and the Virgin Islands. To play Powerball, a participant must purchase a $\$ 2$ ticket, select five numbers from the digits 1 through $69,$ and then select a Powerball number from the digits 1 through $26 .$ To determine the winning numbers for each game, lottery officials draw 5 white balls out a drum of 69 white balls numbered 1 through 69 and 1 red ball out of a drum of 26 red balls numbered 1 through $26 .$ To win the Powerball jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and must also match the number on the red Powerball. The numbers $4-8-19-27-34$ with a Powerball number of 10 provided the record jackpot of $\$ 1.586$ billion (Powerball website). a. How many Powerball lottery outcomes are possible? (Hint: Consider this a two-step random experiment. Select the 5 white ball numbers and then select the 1 red Powerball number.) b. What is the probability that a $\$ 2$ lottery ticket wins the Powerball lottery?

   Powerball Lottery. The Powerball lottery is played twice each week in 44 states, the District of Columbia, and the Virgin Islands. To play Powerball, a participant must purchase a $\$ 2$ ticket, select five numbers from the digits 1 through $69,$ and then select a Powerball number from the digits 1 through $26 .$ To determine the winning numbers for each game, lottery officials draw 5 white balls out a drum of 69 white balls numbered 1 through 69 and 1 red ball out of a drum of 26 red balls numbered 1 through $26 .$ To win the Powerball jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and must also match the number on the red Powerball. The numbers $4-8-19-27-34$ with a Powerball number of 10 provided the record jackpot of $\$ 1.586$ billion (Powerball website).
a. How many Powerball lottery outcomes are possible? (Hint: Consider this a two-step random experiment. Select the 5 white ball numbers and then select the 1 red Powerball number.)
b. What is the probability that a $\$ 2$ lottery ticket wins the Powerball lottery?

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Essentials of Modern Business Statistics

Essentials of Modern Business Statistics

David R. Anderson,… 8th Edition

Chapter 4, Problem 13

Instant Answer

Solved by Expert Nick Johnson

09/16/2022

Step 1

This is a two-step process. First, we select 5 white balls from a pool of 69, and then we select 1 red Powerball from a pool of 26. The number of ways to select 5 white balls from 69 is given by the combination formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where Show more…

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Powerball Lottery. The Powerball lottery is played twice each week in 44 states, the District of Columbia, and the Virgin Islands. To play Powerball, a participant must purchase a $\$ 2$ ticket, select five numbers from the digits 1 through $69,$ and then select a Powerball number from the digits 1 through $26 .$ To determine the winning numbers for each game, lottery officials draw 5 white balls out a drum of 69 white balls numbered 1 through 69 and 1 red ball out of a drum of 26 red balls numbered 1 through $26 .$ To win the Powerball jackpot, a participant's numbers must match the numbers on the 5 white balls in any order and must also match the number on the red Powerball. The numbers $4-8-19-27-34$ with a Powerball number of 10 provided the record jackpot of $\$ 1.586$ billion (Powerball website). a. How many Powerball lottery outcomes are possible? (Hint: Consider this a two-step random experiment. Select the 5 white ball numbers and then select the 1 red Powerball number.) b. What is the probability that a $\$ 2$ lottery ticket wins the Powerball lottery?

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Key Concepts

Combinations

This concept refers to selecting items from a larger set where the order of selection does not matter. In problems like lottery number selection, combinations are used to calculate the number of possible different groups that can be chosen when drawing a subset (such as selecting 5 numbers out of 69). The mathematical technique involves binomial coefficients, often denoted as 'n choose k'.

Multiplication Principle

The multiplication principle is a fundamental counting concept used in probability and combinatorics. It states that if one event can occur in a certain number of ways and a subsequent independent event can occur in another number of ways, then the total number of outcomes for both events occurring is the product of the number of outcomes for each event individually.

Probability Calculation

Probability calculation involves determining the likelihood of an event by dividing the number of favorable outcomes by the total number of possible outcomes. In lottery problems, this concept is applied by finding the ratio of winning number combinations to the total combinations available, providing a measure of how likely a particular event, such as winning a jackpot, is to occur.

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