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Chapter 5: Problem 17

There is a \(1 / 292,201,338\) probability of winning the Powerball lottery jackpot with a single ticket. Assume that you purchase a ticket in each of the next 5200 different Powerball games that are run over the next 50 years. Find the probability of winning the jackpot with at least one of those tickets. Is there a good chance that you would win the jackpot at least once in 50 years?

Short Answer

Expert verified
The probability of winning the Powerball at least once over 50 years of purchasing 5200 tickets is extremely low.

Step by step solution

01

Understand the Problem

Determine the probability of winning the Powerball at least once if 5200 tickets are purchased over 50 years.
02

Calculate the Probability of Losing a Single Game

The probability of losing a single game is the complement of winning. So, the probability of losing is:\[ P(\text{lose}) = 1 - P(\text{win}) = 1 - \frac{1}{292,201,338} \]
03

Calculate the Probability of Losing All Games

The probability of losing all 5200 games is the probability of losing one game raised to the power of the number of games:\[ P(\text{lose all}) = \bigg(1 - \frac{1}{292,201,338}\bigg)^{5200} \]
04

Calculate the Probability of Winning At Least One Game

The probability of winning at least one game is the complement of losing all games:\[ P(\text{win at least one}) = 1 - P(\text{lose all}) \]Using the previous result:\[ P(\text{win at least one}) = 1 - \bigg(1 - \frac{1}{292,201,338}\bigg)^{5200} \]
05

Interpret the Result

Compute the numerical value of the probability expression to determine if there is a good chance of winning at least once over 50 years.
06

Calculation of the Numerical Value

Using a calculator or a software tool, compute the value of:\[ 1 - \bigg(1 - \frac{1}{292,201,338}\bigg)^{5200} \]

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Key Concepts

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

complementary probability
When dealing with the probability of events, it's often helpful to think in terms of complementary events. A complementary event is one that represents all the outcomes that are NOT part of the event we are interested in. For instance, if you want to calculate the probability of winning at least once in a lottery over many tries, it might be easier to first calculate the probability of losing every single time.
large number approximation
When working with extremely large or small numbers, using approximations can make calculations easier. This is particularly useful in probability calculations where precise values are cumbersome.
independent events
In probability, events are independent if the outcome of one event does not affect the outcome of another. This concept is crucial in situations like the lottery, where the result of each draw is completely separate from any other draw.
lottery probability
Understanding the probability of events in a lottery system involves grasping several key concepts. The most important is how improbable winning typically is due to the vast number of possible combinations.

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Most popular questions from this chapter

For 100 births, \(P\) (exactly 56 girls) \(=0.0390\) and \(P(56 \text { or more girls) }=0.136\) Is 56 girls in 100 births a significantly high number of girls? Which probability is relevant to answering that question?

Assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 6.1 per year, as in Example \(I\); and proceed to find the indicated probability. Hurricanes a. Find the probability that in a year, there will be no hurricanes. b. In a 55 -year period, how many years are expected to have no hurricanes? c. How does the result from part (b) compare to the recent period of 55 years in which there were no years without any hurricanes? Does the Poisson distribution work well here?

If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has \(A\) objects of one type (such as lottery numbers you selected), while the remaining \(B\) objects are of the other type (such as lottery numbers you didn't select), and if \(n\) objects are sampled without replacement (such as six drawn lottery numbers), then the probability of getting \(x\) objects of type \(A\) and \(n-x\) objects of type \(B\) is $$P(x)=\frac{A !}{(A-x) ! x !} \cdot \frac{B !}{(B-n+x) !(n-x) !} \div \frac{(A+B) !}{(A+B-n) ! n !}$$ In New Jersey's Pick 6 lottery game, a bettor selects six numbers from 1 to 49 (without repetition), and a winning six-number combination is later randomly selected. Find the probabilities of getting exactly two winning numbers with one ticket. (Hint: Use \(A=6, B=43, n=6\) and \(x=2 .\))

There is a 0.9968 probability that a randomly selected 50 -year-old female lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges \(\$ 226\) for insuring that the female will live through the year. If she does not survive the year, the policy pays out SS0,000 as a death benefit. a. From the perspective of the 50 -year-old female, what are the values corresponding to the two events of surviving the year and not surviving. b. If a 50 -year-old female purchases the policy, what is her expected value? c. Can the insurance company expect to make a profit from many such policies? Why?

Involve finding binomial probabilities, finding parameters, and determining whether values are significantly high or low by using the range rule of thumb and probabilities. Data Set 27 "M\&M Weights" in Appendix B includes data from 100 M\&M candies, and 19 of them are green. Mars, Inc. claims that \(16 \%\) of its plain M\&M candies are green. For the following, assume that the claim of \(16 \%\) is true, and assume that a sample consists of \(100 \mathrm{M} \& \mathrm{Ms}\). a. Use the range rule of thumb to identify the limits separating values that are significantly low and those that are significantly high. Based on the results, is the result of 19 green \(\mathrm{M} \& \mathrm{Ms}\) significantly high? b. Find the probability of exactly 19 green M\&Ms. c. Find the probability of 19 or more green M\&Ms. d. Which probability is relevant for determining whether the result of 19 green \(\mathrm{M} \& \mathrm{Ms}\) is significantly high: the probability from part (b) or part (c)? Based on the relevant probability, is the result of 19 green M\&Ms significantly high? e. What do the results suggest about the \(16 \%\) claim by Mars, Inc.?
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