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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.3.15
Chapter 4, Problem 4.3.15

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)

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Step 1: Identify the type of distribution to use. Since the problem involves finding the probability of the first success (pass completion) occurring on a specific trial, this is a geometric distribution. The probability of success (p) is given as 65.1% or 0.651, and the probability of failure (q) is 1 - p = 0.349.
Step 2: For part (a), use the formula for the geometric distribution: P(X = k) = q^(k-1) * p, where k is the trial number of the first success. Substitute k = 2, p = 0.651, and q = 0.349 into the formula to calculate the probability that the first pass completion occurs on the second pass.
Step 3: For part (b), calculate the probability that the first pass completion occurs on the first or second pass. This is the cumulative probability P(X ≤ 2), which is the sum of the probabilities for k = 1 and k = 2. Use the geometric distribution formula for each value of k and add the results: P(X ≤ 2) = P(X = 1) + P(X = 2).
Step 4: For part (c), calculate the probability that Aaron Rodgers does not complete his first two passes. This is equivalent to the probability of two consecutive failures, which can be expressed as q^2. Substitute q = 0.349 into the formula to find this probability.
Step 5: Determine whether the events are unusual. An event is typically considered unusual if its probability is less than 0.05. Compare the probabilities calculated in parts (a), (b), and (c) to this threshold to assess whether the events are unusual.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. In this context, it is used to find the probability that the first successful pass completion occurs on a specific attempt, such as the second pass. The probability of success remains constant across trials, making it suitable for scenarios like pass completions.
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Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In the given question, it can be applied to find the probability of completing a certain number of passes out of a specified total attempts. This distribution is useful for determining the likelihood of multiple successes in a defined scenario.
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Unusual Events

An event is considered unusual if its probability is significantly low, typically less than 5%. In the context of the question, determining whether the outcomes of Aaron Rodgers' pass completions are unusual involves calculating the probabilities and assessing their significance. This concept helps in understanding the likelihood of specific outcomes in sports statistics and decision-making.
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Related Practice
Textbook Question

Unusual Events In Exercise 19, would it be unusual for a household to have no HD televisions? Explain your reasoning.

Textbook Question

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


P(3) when μ = 6

Textbook Question

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)

Textbook Question

Linear Transformation of a Random Variable In Exercises 41 and 42, use this information about linear transformations. For a random variable x, a new random variable y can be created by applying a linear transformation , where a and b are constants. If the random variable x has mean and standard deviation , then the mean, variance, and standard deviation of y are given by the formulas


The mean annual salary of employees at an office is originally \$46,000. Each employee receives an annual bonus of \$600 and a 3% raise (based on salary). What is the new mean annual salary (including the bonus and raise)?

Textbook Question

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.


Workplace Cleanliness Fifty-seven percent of employees judge their peers by the cleanliness of their workspaces. You randomly select 10 employees and ask them whether they judge their peers by the cleanliness of their workspaces. The random variable represents the number who judge their peers by the cleanliness of their workspaces. (Source: Adecco)

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Textbook Question

Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

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