5. Binomial Distribution & Discrete Random Variables

In Exercises 5–8, find the indicated probability using the Poisson distribution.

P(3) when μ = 6

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is

sigma^2 = q/p^2

Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the grade on an exam worth a total of 100 points.

In your own words, describe the difference between the value of x in a binomial distribution and in the Poisson distribution.

Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Immigration The mean number of people who immigrated to the United States per hour was about 5.5 in April 2021. Find the probability that the number of people who immigrate to the U.S. in a given hour in April 2021 was (a) zero, (b) exactly five, and (c) exactly eight. (Source: U.S. Census Bureau)

Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Pass Completions NFL player Aaron Rodgers completes a pass 65.1% of the time. Find the probability that (a) the first pass he completes is the second pass, (b) the first pass he completes is the first or second pass, and (c) he does not complete his first two passes. (Source: National Football League)

The table lists the number of wireless devices per household in a small town in the United States.

a. Construct a probability distribution.

The table lists the number of wireless devices per household in a small town in the United States.

c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.

In Exercises 3 and 4, (a) construct a probability distribution, and (b) graph the probability distribution using a histogram and describe its shape.

The number of hours students in a college class slept the previous night

In Exercises 7 and 8, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

The number of cell phones per household in a small town

In Exercises 9 and 10, find the expected net gain to the player for one play of the game.

It costs \$25 to bet on a horse race. The horse has a 1/8 chance of winning and a 1/4 chance of placing second or third. You win \$125 if the horse wins and receive your money back if the horse places second or third.

Determine whether the distribution is a probability distribution. If it is not a probability distribution, explain why.

The table shows the ages of students in a freshman orientation course.

a. Construct a probability distribution.

The table shows the ages of students in a freshman orientation course.

b. Graph the probability distribution using a histogram and describe its shape.

The Centers for Disease Control and Prevention (CDC) is required by law to publish a report on assisted reproductive technology (ART). ART includes all fertility treatments in which both the egg and the sperm are used. These procedures generally involve removing eggs from a patient’s ovaries, combining them with sperm in the laboratory, and returning them to the patient’s body or giving them to another patient.

You are helping to prepare a CDC report on young ART patients and select at random 6 ART cycles of patients under 35 years of age for a special review. None of the cycles resulted in a live birth. Your manager feels it is impossible to select at random 10 ART cycles that do not result in a live birth. Use the pie chart at the right and your knowledge of statistics to determine whether your manager is correct.

b. What probability distribution do you think best describes the situation? Do you think the distribution of the number of live births is discrete or continuous? Explain your reasoning.