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.RE.13b

Chapter 4, Problem 4.RE.13b

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.
Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (b) at least three

Verified step by step guidance

Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used when there are a fixed number of independent trials (n), each with two possible outcomes (success or failure), and the probability of success (p) is constant for each trial. Here, n = 8 (number of trials), p = 0.53 (probability of success), and we are looking for the probability of at least 3 successes.

Step 2: Define the complement event. The probability of 'at least 3 successes' can be calculated as 1 minus the probability of 'fewer than 3 successes.' Mathematically, this is expressed as P(X ≥ 3) = 1 - P(X < 3).

Step 3: Break down P(X < 3). This represents the probability of having 0, 1, or 2 successes. Using the binomial probability formula, P(X = k) = (n choose k) * p^k * (1-p)^(n-k), 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). This gives P(X < 3). Use the binomial formula for each term: P(X = k) = (8 choose k) * (0.53)^k * (0.47)^(8-k), where k = 0, 1, 2.

Step 5: Subtract the result of P(X < 3) from 1 to find P(X ≥ 3). This final step gives the probability of at least 3 successes. If using technology or a binomial probability calculator, input n = 8, p = 0.53, and calculate P(X ≥ 3) directly.

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Key Concepts

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

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the trials are the responses of the eight randomly selected U.S. adults regarding their support for landing an astronaut on Mars, with a success defined as a 'yes' response.

Probability Calculation

To find the probability of a specific number of successes in a binomial distribution, we use the binomial probability formula: 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, and p is the probability of success. For this question, we need to calculate the probability of at least three supporters, which involves summing the probabilities of getting three, four, five, six, seven, or eight supporters.

Cumulative Probability

Cumulative probability refers to the probability of obtaining a value less than or equal to a certain number in a distribution. In this case, to find the probability of at least three supporters, we can either calculate the probabilities for three or more supporters directly or use the complement rule by finding the cumulative probability of having fewer than three supporters and subtracting it from one.

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Related Practice

Textbook Question

In Exercises 21–26, 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.

Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (b) the fourth or fifth person selected.

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Seventy-two percent of U.S. civilian employees have access to medical care benefits. You randomly select nine civilian employees. Find the probability that the number who have access to medical care benefits is (b) at least six

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Textbook Question

In Exercises 11 and 12, determine whether the experiment is a binomial experiment. If it is, identify a success; specify the values of n, p, and q; and list the possible values of the random variable x. If it is not a binomial experiment, explain why.

A fair coin is tossed repeatedly until 15 heads are obtained. The random variable x counts the number of tosses.

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Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

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Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

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