Ch. 5 - Discrete Probability Distributions

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 5 - Discrete Probability Distributions

Problem 5.2.27c

Chapter 5, Problem 5.2.27c

In Exercises 25–28, find the probabilities and answer the questions.

Internet Voting Based on a Consumer Reports survey, 39% of likely voters would be willing to vote by Internet instead of the in-person traditional method of voting. For each of the following, assume that 15 likely voters are randomly selected.

c. Find the probability that at least one of the selected likely voters would do Internet voting.

Verified step by step guidance

Step 1: Recognize that this is a binomial probability problem. The random variable represents the number of likely voters willing to vote by Internet, and the probability of success (willing to vote by Internet) is 0.39. The number of trials is 15.

Step 2: To find the probability that at least one voter would do Internet voting, use the complement rule. The complement of 'at least one' is 'none,' meaning no voters are willing to vote by Internet. Calculate the probability of no successes (P(X = 0)).

Step 3: Use the binomial probability formula to calculate P(X = 0): P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and (n choose k) is the binomial coefficient. For P(X = 0), k = 0, n = 15, and p = 0.39.

Step 4: Compute the complement probability: P(at least one voter) = 1 - P(X = 0). This step ensures you find the probability of at least one voter willing to vote by Internet.

Step 5: Interpret the result in the context of the problem. The calculated probability represents the likelihood that at least one of the 15 selected likely voters would be willing to vote by Internet.

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Key Concepts

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

Binomial Probability

Binomial probability refers to the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the success is defined as a voter choosing to vote via the Internet. The formula for binomial probability can be used to calculate the probability of exactly k successes in n trials.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of the event not occurring. In this case, to find the probability that at least one voter would choose Internet voting, it is often easier to first calculate the probability that none of the voters would choose this method and then subtract that value from one.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. For this problem, the distribution of the number of voters choosing Internet voting can be modeled using a binomial distribution, where the parameters are the number of trials (15 voters) and the probability of success (39%). This distribution helps in calculating various probabilities related to the voting preferences.

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Calculating Probabilities in a Binomial Distribution

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Textbook Question

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.

Gender Selection Assume that the groups consist of 36 couples.

c. Is the result of 26 girls a result that is significantly high? What does it suggest about the effectiveness of the XSORT method?

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Textbook Question

Using Probabilities for Significant Events

c. Which probability is relevant for determining whether 3 is a significantly high number of matches: the result from part (a) or part (b)?

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Textbook Question

Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.

Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.

c. Based on the preceding results, what is the probability of getting exactly one correct answer when three guesses are made?

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Textbook Question

Salary Negotiations In a Jobvite survey, 2287 adult workers were randomly selected and asked about salary negotiations.

b. Among those who negotiated salary, 84% received higher pay. How many received higher pay?

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Textbook Question

Planets The planets of the solar system have the numbers of moons listed below in order from the sun. (Pluto is not included because it was uninvited from the solar system party in 2006.) Include appropriate units whenever relevant.

0 0 1 2 17 28 21 8

e. Find the standard deviation.

f. Find the variance.

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Textbook Question

Salary Negotiations In a Jobvite survey, 2287 adult workers were randomly selected and asked about salary negotiations.

a. 29% of the respondents reported that they negotiated salary at their latest job. What is the number of respondents who reported that they negotiated salary?

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