Ch. 4 - Discrete Probability Distributions

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 4 - Discrete Probability Distributions

Problem 4.2.37c

Chapter 4, Problem 4.2.37c

Unusual Events In Exercises 37 and 38, find the indicated probabilities. Then determine if the event is unusual. Explain your reasoning.

Rock-Paper-Scissors The probability of winning a game of rock-paper-scissors is 1/3. You play nine games of rock-paper-scissors. Find the probability that the number of games you win is (c) less than two.

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Step 1: Recognize that this is a binomial probability problem because there are a fixed number of trials (9 games), two possible outcomes for each trial (win or not win), and the probability of success (winning) is constant at 1/3.

Step 2: Define the random variable X as the number of games won. X follows a binomial distribution with parameters n = 9 (number of trials) and p = 1/3 (probability of success). The probability mass function for a binomial distribution is given by: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'k' is the number of successes.

Step 3: To find the probability that the number of games won is less than 2, calculate P(X < 2). This is equivalent to P(X = 0) + P(X = 1). Use the binomial probability formula for each term: P(X = 0) = (9 choose 0) * (1/3)^0 * (2/3)^9 and P(X = 1) = (9 choose 1) * (1/3)^1 * (2/3)^8.

Step 4: Compute the binomial coefficients (n choose k) for each term. For P(X = 0), (9 choose 0) = 1. For P(X = 1), (9 choose 1) = 9. Substitute these values into the respective probability expressions.

Step 5: Add the probabilities P(X = 0) and P(X = 1) to get P(X < 2). Finally, compare the result to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In the context of games like rock-paper-scissors, the probability of winning a single game is 1/3, meaning that in a large number of games, you would expect to win about one-third of the time. Understanding probability is essential for calculating the chances of winning multiple games.

Binomial Distribution

The binomial distribution is a statistical distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, playing nine games of rock-paper-scissors can be modeled as a binomial distribution where the number of trials is 9 and the probability of winning each trial is 1/3. This concept is crucial for calculating the probability of winning fewer than two games.

Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In the context of the question, after calculating the probability of winning less than two games, you would compare this probability to the threshold of 5% to determine if the outcome is unusual. This concept helps in assessing the significance of the results in statistical analysis.

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Related Practice

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.

Oil Tankers In the month of June 2021, 240 oil tankers stop at a port city. No oil tanker visits more than once. Find the probability that the number of oil tankers that stop on any given day in June is (c) more than eight.

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Textbook Question

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.

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Textbook Question

Living Donor Transplants The mean number of organ transplants from living donors performed per day in the United States in 2020 was about 16. Find the probability that the number of organ transplants from living donors performed on any given day is (c) no more than 10. (Source: Organ Procurement and Transplantation Network)

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Textbook Question

Living Donor Transplants The mean number of organ transplants from living donors performed per day in the United States in 2020 was about 16. Find the probability that the number of organ transplants from living donors performed on any given day is (b) at least eight (Source: Organ Procurement and Transplantation Network)

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Textbook Question

Hypergeometric Distribution Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and failures, the probability of selecting a sample of size that has successes and failures is given by

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In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Use the above formula to find the probability that (c) two microchips are defective and one is not defective.

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Textbook Question

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (c) from one to three HD televisions,

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