Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.


Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.


a. Use the multiplication rule to find the probability that the first two guesses are wrong and the third is correct. That is, find P(WWC), where W denotes a wrong answer and C denotes a correct answer.

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.

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.

In a Pew Research Center survey, 3930 subjects were asked if they have ever fired a gun, and the responses consist of “yes” or “no.”

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.

LOL In a U.S. Cellular survey of 500 smartphone users, subjects are asked if they find abbreviations (such as LOL or BFF) annoying, and each response was recorded as “yes,” “no,” or “not sure.”

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.

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

a. Find the mean and standard deviation for the numbers of girls in groups of 36 births.

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

c. Is the result of 26 girls a result that is significantly high? What does it suggest about the effectiveness of the XSORT method?

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.

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 40 first lines for Democrats significantly high?