Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.


HIV It is estimated that in the United States, the proportion of people infected with the human immunodeficiency virus (HIV) is 0.00343. In tests for HIV, blood samples from 50 different people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

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?

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.

b. If you play this game once every day, find the mean number of wins in years with exactly 365 days.

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.

c. If you play this game once every day, find the probability of no wins in 365 days.

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?

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.

b. If one of the overturned calls is randomly selected, what is the probability that the challenge was made by a woman?

Notation Assume that we want to find the probability that when five speaking characters in movies are randomly selected, exactly two of them are females. Also assume that when randomly selecting a speaking character in a movie, the probability of getting a female is 0.331. Identify the values of n, x, p, and q.

Independent 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?

In Exercises 5–12, determine whether the given procedure results in a binomial distribution or a distribution that can be treated as binomial (by applying the 5% guideline for cumbersome calculations). For those that are not binomial and cannot be treated as binomial, identify at least one requirement that is not satisfied.

Pew Survey In a Pew Research Center survey of 3930 subjects, the ages of the respondents are recorded.