5. Binomial Distribution & Discrete Random Variables

Which of the following expressions correctly represents the mean of a binomial distribution with parameters $n$ and $p$?

Which of the following is not an example of a binomial distribution?

Finding Binomial Probabilities In Exercises 19–26, find the indicated probabilities. If convenient, use technology or Table 2 in Appendix B.

Penalty Kicks Argentine soccer player Lionel Messi converts 78% of his penalty kicks. Suppose Messi takes six penalty kicks next season. Find the probability that the number he converts is (a) exactly six, (b) at most three, and (c) more than three. (Source: Transfermarkt)

Which of the following is not a requirement of the binomial probability distribution?

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.

e. If one of the challenges is randomly selected, find the probability that it was made by a man, given that the challenge was upheld with an overturned call.

In a binomial experiment with $n=10$ trials and probability of success $p=0.10$, what is the expected value (mean) of the number of successes $\left(E\right(X)=np)$?

Which of the following is not a requirement of the binomial probability distribution?

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.

b. Find the probability of exactly 40 first lines for Democrats.

For the binomial distribution with parameters $n$ and $p$, which of the following expressions gives the probability of exactly $k$ successes in $n$ independent trials?

Which of the following is a necessary condition for using the $binomial$ distribution?

A gardener plants 8 seeds, and each has a 65% probability of germinating successfully. Find the probability that less than 4 seeds germinate successfully.

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (d) at most two HD televisions.

Identifying and Understanding Binomial Experiments In Exercises 15–18, determine whether the experiment is a binomial experiment. If it is, identify a success; specify the values of n, p, and q; and list the possible values of the random variable x. If it is not a binomial experiment, explain why.

Video Games A survey found that 29% of gamers own a virtual reality (VR) device. Ten gamers are randomly selected. The random variable represents the number who own a VR device. (Source: Entertainment Software Association)

Which of the following graphs best represents the probability mass function of a binomial distribution with $n=20$ and $p=0.25$?

Driving AgeAccording to a Gallup poll, 60% of U.S. women 18 years old or older stated that the minimum driving age should be 18. In a random sample of 15 U.S. women 18 years old or older, find the probability that: 

d. Between 7 and 12, inclusive, believe that the minimum driving age should be 18.