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Chapter 5: Problem 84

The president of a company specializing in public opinion surveys claims that approximately \(70 \%\) of all people to whom the agency sends questionnaires respond by filling out and returning the questionnaire. Twenty such questionnaires are sent out, and assume that the president's claim is correct. a. What is the probability that exactly ten of the questionnaires are filled out and returned? b. What is the probability that at least 12 of the questionnaires are filled out and returned? c. What is the probability that at most ten of the questionnaires are filled out and returned?

Short Answer

Expert verified
Answer: The probabilities are as follows: a) Probability that exactly 10 questionnaires are returned: 鈮� 0.0364 b) Probability that at least 12 questionnaires are returned: 鈮� 0.9873 c) Probability that at most 10 questionnaires are returned: 鈮� 0.0537

Step by step solution

01

a. Probability that exactly 10 questionnaires are returned

For a binomial distribution, the probability of having exactly k successes in n trials is given by: P(X = k) = C(n, k) * p^k * q^(n-k) In our case, we're finding the probability when k = 10: P(X = 10) = C(20, 10) * (0.70)^{10} * (0.30)^{(20-10)} Use the combination formula to find C(20, 10), which is equal to 184756 and compute the probability: P(X = 10) = 184756 * (0.70)^{10} * (0.30)^{10} 鈮� 0.0364
02

b. Probability that at least 12 questionnaires are returned

To find the probability that at least 12 questionnaires are returned, we need to calculate the probability that 12 or more questionnaires are returned, which includes the probabilities of 12, 13, 14... (up to 20) returned questionnaires. We can do this by finding the complement probability, which is the probability that fewer than 12 questionnaires are returned. So, we have: P(X 鈮� 12) = 1 - P(X < 12) P(X 鈮� 12) = 1 - 危[P(X = k)] for k = 0 to 11 Now, we sum up the probabilities for k from 0 to 11 and subtract it from 1 to find the probability that at least 12 questionnaires are returned: P(X 鈮� 12) 鈮� 1 - 9.69 * 10^{-4} - = 7.81 * 10^{-3} = 9.99 * 10^{-4}... = 0.9873
03

c. Probability that at most 10 questionnaires are returned

To find the probability that at most 10 questionnaires are returned, we can sum up the probabilities for k = 0 to 10: P(X <= 10) = 危[P(X = k)] for k = 0 to 10 Now, we sum up the probabilities for k from 0 to 10 to find the probability that at most 10 questionnaires are returned: P(X 鈮� 10) 鈮� 4.02 * 10^{-13} + 1.34 * 10^{-10} + 2.05 * 10^{-8} ... = 0.0537 So, the probabilities we have calculated are: a) P(X = 10) 鈮� 0.0364 b) P(X 鈮� 12) 鈮� 0.9873 c) P(X 鈮� 10) 鈮� 0.0537

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Probability is a foundational concept in statistics and mathematics. It expresses how likely an event is to occur, ranging from 0 to 1. A probability of 0 means the event will not happen, while a probability of 1 means it will happen for sure.
To determine the probability of an event in a repeated trial process, like mailing surveys, understanding the Binomial Distribution is essential. In our exercise, we used this distribution to find probabilities of specific outcomes - such as the number of surveys returned.
For example:
  • When calculating the chance of exactly 10 surveys being returned out of 20 mailed, the probability formula for a binomial distribution is applied: \[ P(X = k) = C(n, k) \times p^k \times q^{(n-k)} \]
  • Here, \( p \) is the probability of success (survey returned), and \( q \) is its complement (survey not returned). In this case, \( p = 0.70 \) and \( q = 0.30 \).
Probability helps us understand expectations in real-world events where outcomes are uncertain.
Combinatorics
Combinatorics is a field of mathematics focusing on counting, arrangement, and combination of elements within sets. It's crucial when dealing with scenarios where order doesn't matter.
In the context of a binomial distribution, combinatorics comes into play when calculating combinations, denoted as \( C(n, k) \) or "n choose k." This represents the number of ways to choose \( k \) successes (e.g., returned surveys) from \( n \) trials (total mailed surveys), disregarding order.
Using the formula for combinations:\[C(n, k) = \frac{n!}{k!(n-k)!}\]
  • \( n! \) ("n factorial") is the product of all positive integers up to \( n \).
  • For our problem, \( C(20, 10) \) calculated how many ways 10 returns could occur out of 20 distributed surveys, resulting in 184756 different ways.
Combinatorics is essential for accurately determining probabilities in binomial scenarios.
Successes and Trials
"Successes" and "trials" are terms frequently used in the binomial context to describe the nature of an experiment.
  • **Trials:** Each individual attempt or choice in the experiment, like sending one survey.
  • **Successes:** The desired result for each trial, like receiving a filled survey back.
In a binomial probability setting, we're often interested in the number of successes across a series of trials. This number is the random variable, which can adopt values from 0 up to the total number of trials.
Let's break it down with our survey problem:
  • You have 20 trials (surveys sent out).
  • Success in each trial occurs if the survey is returned.
  • The probability of success for each trial is the same, \( p = 0.70 \).
  • You can calculate the probability for any number of successes, like the probability of getting exactly 10 successes, thanks to each trial being independent and having two possible outcomes.

Understanding the number of successes across trials is the heart of calculating binomial probabilities.

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Most popular questions from this chapter

Consider the medical payment problem in Exercise 5.27 in a more realistic setting. Of all patients admitted to a medical clinic, \(30 \%\) fail to pay their bills and the debts are eventually forgiven. If the clinic treats 2000 different patients over a period of 1 year, what is the mean (expected) number of debts that have to be forgiven? If \(x\) is the number of forgiven debts in the group of 2000 patients, find the variance and standard deviation of \(x\). What can you say about the probability that \(x\) will exceed \(700 ?\) (HINT: Use the values of \(\mu\) and \(\sigma,\) along with Tchebysheff's Theorem, to answer this question.)

Find the mean and standard deviation for a binomial distribution with \(n=100\) and these values of \(p:\) a. \(p=.01\) b. \(p=.9\) c. \(p=.3\) d. \(p=.7\) e. \(p=.5\)

Fast Food and Gas Stations Forty percent of all Americans who travel by car look for gas stations and food outlets that are close to or visible from the highway. Suppose a random sample of \(n=25\) Americans who travel by car are asked how they determine where to stop for food and gas. Let \(x\) be the number in the sample who respond that they look for gas stations and food outlets that are close to or visible from the highway. a. What are the mean and variance of \(x ?\) b. Calculate the interval \(\mu \pm 2 \sigma\). What values of the binomial random variable \(x\) fall into this interval? c. Find \(P(6 \leq x \leq 14)\). How does this compare with the fraction in the interval \(\mu \pm 2 \sigma\) for any distribution? For mound-shaped distributions?

A new surgical procedure is said to be successful \(80 \%\) of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another. What are the probabilities of these events? a. All five operations are successful. b. Exactly four are successful. c. Less than two are successful.

Suppose that \(10 \%\) of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected and checked for whitefly. a. What is the average number of fields sampled that are infested with whitefly? b. Within what limits would you expect to find the number of infested fields, with probability approximately \(95 \% ?\) c. What might you conclude if you found that \(x=25\) fields were infested? Is it possible that one of the characteristics of a binomial experiment is not satisfied in this experiment? Explain.
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