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The Powerball lottery is played twice each week in 28 states, the Virgin Islands, and the District of Columbia. To play Powerball a participant must purchase a ticket and then select five numbers from the digits 1 through 55 and a Powerball number from the digits 1 through 42\. To determine the winning numbers for each game, lottery officials draw five white balls out of a drum with 55 white balls, and one red ball out of a drum with 42 red balls. To win the jackpot, a participant's numbers must match the numbers on the five white balls in any order and the number on the red Powerball. Eight coworkers at the ConAgra Foods plant in Lincoln, Nebraska, claimed the record \(\$ 365\) million jackpot on February \(18,2006,\) by matching the numbers \(15-17-43-44-49\) and the Powerball number \(29 .\) A variety of other cash prizes are awarded each time the game is played. For instance, a prize of \(\$ 200,000\) is paid if the participant's five numbers match the numbers on the five white balls (http://www.powerball.com, March 19, 2006). a. Compute the number of ways the first five numbers can be selected. b. What is the probability of winning a prize of \(\$ 200,000\) by matching the numbers on the five white balls? c. What is the probability of winning the Powerball jackpot?

Step by step solution

01

Calculating the Number of Ways to Select Five White Balls

To find the number of ways to select five numbers from the white balls, we use the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, \( n = 55 \) (since there are 55 white balls) and \( k = 5 \). Thus, the combination is:\[ \binom{55}{5} = \frac{55 \times 54 \times 53 \times 52 \times 51}{5 \times 4 \times 3 \times 2 \times 1} = 3,478,761 \]So there are 3,478,761 ways to select the first five white ball numbers.

02

Calculating Probability of Matching Five White Balls

The probability of matching the numbers on the five white balls is the number of favorable outcomes divided by the total possible outcomes.The number of favorable outcomes is 1 (since you're choosing the exact winning numbers). Thus, \[ \text{Probability} = \frac{1}{\binom{55}{5}} = \frac{1}{3,478,761} \]

03

Calculating Probability of Winning Powerball Jackpot

To win the Powerball jackpot, you need to match five white ball numbers and the Powerball number.We already calculated the number of ways to select the five white balls as 3,478,761. There are 42 possible red Powerball numbers.Thus, the total combinations are \( 3,478,761 \times 42 \).The probability of winning the jackpot is the reciprocal of the total combinations which is:\[ \text{Probability} = \frac{1}{3,478,761 \times 42} = \frac{1}{146,107,962} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics

Combinatorics is a field of mathematics that deals with counting, or to be more specific, the counting of possible arrangements in a set of objects. When you play the lottery, combinatorics helps to understand how many different ways you can choose numbers. This relates closely to combinations and permutations.
In lottery problems, especially like Powerball, combinations are used instead of permutations because the order of the numbers doesn't matter. For example, choosing the numbers 3, 5, and 2 is the same as choosing the numbers 2, 3, and 5. This is why we use the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose. Factorials (!) are used, which means multiplying a series of descending numbers. For instance, 5! = 5 x 4 x 3 x 2 x 1.

• Combination focuses on how groups can be formed from a larger pool.
• Understanding combinations helps in calculating probabilities in games of chance like lottery draws.

Lottery

A lottery is a form of gambling where numbers are drawn for a prize. In many lotteries, players select numbers from a fixed range, such as 1 through 55 in the Powerball, and then a separate number known as the "Powerball" is drawn from a different set of numbers.
The aim is often to match your selected numbers with those drawn by the lottery to win prizes. The numbers can be chosen manually or generated randomly. The excitement of lotteries comes from the small probability of winning big and the typically large prizes at stake. Considerations when participating in lotteries include:

• Lotteries are purely chance-based with no guaranteed strategies to win.
• The odds can greatly vary, influencing potential payout scales.
• Lotteries often contribute a portion of sales to charity or public services.

Jackpot

The term 'jackpot' usually refers to the top prize in a lottery—like the $365 million Powerball jackpot won by eight coworkers in 2006. Winning the jackpot involves matching all of the drawn numbers plus the separate Powerball number. This makes it the hardest prize to win.
The jackpot grows over time if no one wins, as the money that would have been awarded is added to the next drawing's jackpot, often leading to extreme build-ups in prize totals.

• To win bigger jackpots, players must match an increasing number of draw elements.
• Jackpot sizes typically depend on ticket sales and periodic no-win rollovers.
• A single jackpot winner may choose a lump sum payout or an annuity over time.

Statistics

Statistics involves the analysis of data to make predictions or understand patterns. In the lottery, statisticians use probability to deduce how likely it is to win a certain prize. The probability of winning the Powerball jackpot is extremely low. To understand this:

• Winning the jackpot requires one ticket to match all numbers drawn. This probability can be as low as 1 in 146 million.
• Statistics help players understand their odds, sometimes dissuading over-confidence in their chances of winning huge prizes.
• By computing probabilities and expected values, players can make informed decisions about participating.

Statistics highlight the importance of understanding risks and making calculated decisions in gambling scenarios like lotteries.