5. Binomial Distribution & Discrete Random Variables

Which of the following scenarios describes a binomial experiment?

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 If none of the ten workers tests positive for illegal drugs, is that a significantly low result?

LeBron scores $70\%$ of his free throws. If he shoots $30$ free throws, which of the following expressions gives the probability that he makes exactly $21$ of them?

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.

Basketball A’ja Wilson, the 2020 WNBA Most Valuable Player, makes a free throw shot about 78% of the time. The random variable represents the number of free throws that she makes on eight attempts. (Source: Women’s National Basketball Association)

Suppose a binomial probability experiment is conducted with $n=10$ trials and probability of success $p=0.3$. What is the probability of exactly $4$ successes?

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.

College Acceptance Pennsylvania State University accepts 49% of applicants. You randomly select seven Pennsylvania State University applicants. The random variable represents the number who are accepted. (Source: US News & World Report)

Happiness In a survey sponsored by Coca-Cola, subjects were asked what contributes most to their happiness, and the table summarizes their responses. Does the table represent a probability distribution? Explain.

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.

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 152 yellow peas either significantly low or significantly high?

In Exercises 17 and 18, (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.

Seventy-two percent of U.S adults have read a book in any format in the past year. You randomly select five U.S adults and ask them whether they have read a book in any format in the past year. The random variable represents the number of adults who have read a book in any format in the past year. (Source: Pew Research)

In a binomial distribution, how does increasing the value of $n$ (the number of trials) affect the shape of the binomial probability histogram?

Which of the following statements is true for both the binomial and Poisson distributions?

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: 

e. In a random sample of 200 U.S. women 18 years old or older, what is the expected number who believe that the minimum driving age should be 18? What is the standard deviation?

In a binomial experiment, how many possible outcomes are there for a single trial?

Which of the following is a criterion for a binomial probability experiment?

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