Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 3 - Probability

Problem 3.2.26c

Chapter 3, Problem 3.2.26c

Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
26. Worst President In a sample of 1500 adult U.S. citizens, 690 said that Donald Trump was the worst president in U.S. history. Three adult U.S. citizens are selected at random.
(Adapted from YouGov)
c. Find the probability that at most two of the three adult U.S. citizens say that Donald Trump was the worst president in U.S. history.

Verified step by step guidance

Step 1: Define the problem using the given data. Let the probability that an adult U.S. citizen says Donald Trump was the worst president be p = 690/1500. Calculate this probability.

Step 2: Recognize that this is a binomial probability problem because there are a fixed number of trials (n = 3), two possible outcomes (success or failure), and the probability of success (p) is constant for each trial.

Step 3: Use the binomial probability formula to calculate the probability of at most two successes. The formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient. Here, you need to calculate P(X = 0), P(X = 1), and P(X = 2).

Step 4: Add the probabilities for P(X = 0), P(X = 1), and P(X = 2) to find the total probability of at most two successes. This is expressed as P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2).

Step 5: Simplify each term in the summation using the binomial coefficient and the given probability p. Then, sum the results to find the final probability of at most two successes.

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Key Concepts

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

Multiplication Rule of Probability

The Multiplication Rule of Probability states that the probability of two or more independent events occurring together is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple citizens expressing the same opinion about Donald Trump being the worst president, by multiplying the probabilities of each citizen's response.

Binomial Distribution

The Binomial Distribution is a probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this scenario, it can be used to determine the probability of 0, 1, or 2 citizens out of 3 selecting Trump as the worst president, given the proportion of the population that holds this view.

At Most Probability

The term 'at most' refers to the maximum number of successes in a given scenario. In this question, finding the probability that at most two out of three citizens say Trump is the worst president involves calculating the probabilities for 0, 1, and 2 successes and summing these probabilities to get the total likelihood.

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Introduction to Probability

Related Practice

Textbook Question

2. Determine whether each number could represent the probability of an event. Explain your reasoning. c. 2.3

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Textbook Question

"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.

24. Knowing a Person Who Was Murdered In a sample of 11,771 children ages 2 to 17, 8% have lost a friend or relative to murder. Four children are selected at random. (Adapted from University of New Hampshire)

c. Find the probability that at least one of the four has lost a friend or relative to murder."

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Textbook Question

81. Genetics A Punnett square is a diagram that shows all possible gene combinations in a cross of parents whose genes are known. When two pink snapdragon flowers (RW) are crossed, there are four equally likely possible outcomes for the genetic makeup of the offspring: red (RR), pink (RW), pink (WR), and white (WW), as shown in the Punnett square at the left. When two pink snapdragons are crossed, what is the probability that the offspring will be (c) white?

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Textbook Question

Politics The responses of 1500 U.S. adults to a survey that asked them to state their own political viewpoints are shown in the Pareto chart. Find the probability of each event.(Adapted from YouGov)

d. Randomly selecting a person from the sample who is either unsure or moderate

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Textbook Question

Mental Health A survey asks 4805 parents the severity of the mental issues they experienced from the coronavirus pandemic. The results are shown in the table. A parent is randomly selected from the sample. Find the probability of each event. (Adapted from Kaiser Family Foundation)

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c. The parent did not have major mental health issues or is a mother.

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Textbook Question

23. Engineering Degrees The table shows the numbers of male and female students in the U.S. who received B.S. degrees in engineering in a recent year. A student earning a B.S. degree in engineering during that year is selected at random. Find the probability of each event.

(Source: National Center for Educational Statistics)

c. The student is not female or did not receive a mechanical engineering degree.

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