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 4.2.37a

Chapter 4, Problem 4.2.37a

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 (a) exactly five

Verified step by step guidance

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

Step 2: Use the binomial probability formula to calculate the probability of winning exactly 5 games. The formula is: P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)), where n is the number of trials (9), k is the number of successes (5), and p is the probability of success (1/3).

Step 3: Calculate the binomial coefficient (n choose k), which is represented as C(n, k) = n! / (k! * (n-k)!). For this problem, calculate C(9, 5).

Step 4: Substitute the values into the binomial probability formula. Use p = 1/3, n = 9, and k = 5. The formula becomes: P(X = 5) = C(9, 5) * (1/3)^5 * (2/3)^4.

Step 5: After calculating the probability, compare it 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. Explain your reasoning based on the calculated probability.

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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 if you play many games, you can expect to win about one-third of the time. Understanding probability is essential for calculating the chances of winning a specific number of 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, winning a game of rock-paper-scissors can be modeled as a binomial distribution where the number of trials is nine (the games played) and the probability of success (winning) is 1/3. This concept is crucial for determining the probability of winning exactly five 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 exactly five games, you would compare this probability to the threshold of 5% to determine if winning five games is an unusual outcome. This concept helps in making inferences about the likelihood of certain results occurring in probabilistic scenarios.

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Textbook Question

Manufacturing An assembly line produces 10,000 automobile parts. Twenty percent of the parts are defective. An inspector randomly selects 10 of the parts

a. Use the Multiplication Rule (discussed in Section 3.2) to find the probability that none of the selected parts are defective. (Note that the events are dependent.)

Textbook Question

Finding Probabilities Use the probability distribution you made in Exercise 19 to find the probability of randomly selecting a household that has (b) two or more HD televisions

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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 (a) exactly eight

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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 (a) all three microchips are not defective

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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 (a) exactly 12 (Source: Organ Procurement and Transplantation Network)

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Textbook Question

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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 (b) one microchip is defective and two are not defective

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