BackIndependent Events Again assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331, as in Exercise 1. If we want to find the probability of 20 females when 50 different speaking characters are randomly selected from a population of 1500 speaking characters, are the 50 selections independent? Using the 5% guideline for cumbersome calculations, can they be treated as being independent?
Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and failures, the probability of selecting a sample of size that has successes and failures is given by
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In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (c) two microchips are defective and one is not defective.
In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).
Hybrids Assume that offspring peas are randomly selected in groups of 16.
c. Is a result of 7 peas with green pods a result that is significantly low? Why or why not?
In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.
Workplace Drug Testing Find the probability that at least one of the ten workers tests positive for illegal drugs.
Constructing and Graphing Binomial Distributions In Exercises 27–30, (a) construct a binomial distribution, (b) graph the binomial distribution using a histogram and describe its shape, and (c) identify any values of the random variable x that you would consider unusual. Explain your reasoning.
Workplace Cleanliness Fifty-seven percent of employees judge their peers by the cleanliness of their workspaces. You randomly select 10 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number who judge their peers by the cleanliness of their workspaces. (Source: Adecco)
Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.
Responsible Consumption Forty-five percent of consumers say it is important that the clothing they buy is made without child labor. You randomly select 16 consumers. Find the probability that the number of of consumers who say it is important that the clothing they buy is made without child labor is (a) e
Tennis Challenge In a recent U.S. Open tennis tournament, there were 945 challenges made by singles players, and 255 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.
c. If two different challenges are randomly selected without replacement, find the probability that they both resulted in an overturned call.
Biometric Security In a USA Today survey of 510 people, 270 (or 53%) said that we should replace passwords with biometric security, such as fingerprints. Use the following probabilities related to determining whether the result of 270 is significantly high (assuming the true rate is 50%). Is 270 significantly high? What should be concluded about the claim that the majority of the population says that we should replace passwords with biometric security? Explain.
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Forty-nine percent of U.S. adults think that human activity such as burning fossil fuels contributes a great deal to climate change. You randomly select 25 U.S. adults. Find the probability that the number who think that human activity contributes a great deal to climate change is (b) between 8 and 11, inclusive,
Kentucky Pick 4 In the Kentucky Pick 4 lottery game, you can pay \$1 for a “straight” bet in which you select four digits with repetition allowed. If you buy only one ticket and win, your prize is \$2500.
a. If you buy one ticket, what is the probability of winning?
An online magazine finds that the mean number of typographical errors per page is five. Find the probability that the number of typographical errors found on any given page is (a) exactly five, (b) less than five, and (c) exactly zero.
In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.
Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (a) exactly three
Tennis Challenge In a recent U.S. Open tennis tournament, there were 945 challenges made by singles players, and 255 of them resulted in referee calls that were overturned. The accompanying table lists the results by gender.
a. If 1 of the 945 challenges is randomly selected, what is the probability that it resulted in an overturned call?
Internet Traffic Data Set 27 “Internet Traffic” includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?
In Exercises 25–28, find the probabilities and answer the questions.
Internet Voting Based on a Consumer Reports survey, 39% of likely voters would be willing to vote by Internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected.
a. What is the probability that exactly 12 of those selected would do Internet voting?