Question

A public opinion research firm claims that approximately \(70 \%\) of those sent questionnaires respond by 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 10 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 10 of the questionnaires are filled out and returned?

Short answer

Answer: The probabilities are approximately: - 3.68% for exactly 10 questionnaires returned - 95.98% for at least 12 questionnaires returned - 5.74% for at most 10 questionnaires returned

Step-by-step solution

Identify the given parameters

We are given the following information: - The probability of returning a questionnaire is \(p = 0.7\) or \(70\%\) - The number of questionnaires sent out is \(n = 20\) - We are asked to find probabilities for 3 cases: exactly 10, at least 12, and at most 10 questionnaires being returned

Apply the binomial probability formula

The binomial probability formula is as follows: \(P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\) Here, \(P(X=k)\) is the probability of getting k successes (questionnaires returned) in n trials, \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success.

Calculate the probability for each case

a. Exactly 10 questionnaires are returned: \(P(X=10) = \binom{20}{10} \cdot (0.7)^{10} \cdot (0.3)^{10} \approx 0.0368\) b. At least 12 questionnaires are returned: Since we need to find the probability of getting 12 or more questionnaires, we'll calculate the probability for \(k=12, 13, 14, \dots, 20\) and add them together: \(P(X \geq 12) = \sum_{k=12}^{20} \binom{20}{k} \cdot (0.7)^{k} \cdot (0.3)^{20-k} \approx 0.9598\) c. At most 10 questionnaires are returned: Similar to b, we'll calculate the probability for \(k=0, 1, 2, \dots, 10\) and add them together: \(P(X \leq 10) = \sum_{k=0}^{10} \binom{20}{k} \cdot (0.7)^{k} \cdot (0.3)^{20-k} \approx 0.0574\) To summarize the results: a. The probability that exactly 10 questionnaires are returned is approximately \(0.0368\) or \(3.68\%\). b. The probability that at least 12 questionnaires are returned is approximately \(0.9598\) or \(95.98\%\). c. The probability that at most 10 questionnaires are returned is approximately \(0.0574\) or \(5.74\%\).

Key concepts

Probability

Probability is a way to measure how likely an event is to happen. In this exercise, we're finding the probability of different numbers of questionnaires being returned. The concept of probability helps us predict outcomes and analyze risks in uncertain situations.

Probability values range between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In the context of our exercise, when we say that the probability of returning a questionnaire is 0.7, it suggests that there is a 70% chance that any given questionnaire will be returned.

To solve such problems, identifying if they adhere to a specific probability distribution helps to apply the correct formulas and methods to get an accurate result.

Binomial Probability Formula

The binomial probability formula is crucial when dealing with problems like this one, where each trial (sending a questionnaire) has two possible outcomes: success (returned) or failure (not returned).

The formula is: \(P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\)

Here's what each term means:

• \(\binom{n}{k}\) denotes the binomial coefficient, representing the number of ways to choose \(k\) successes from \(n\) trials.


• \(p^k\) is the probability that all \(k\) successes occur.


• \((1-p)^{n-k}\) is the probability that the remaining \(n-k\) trials result in failure.

Thus, the formula allows us to compute the probability of observing exactly \(k\) successes in \(n\) trials, given the probability \(p\) of one success.

Success and Failure

In a binomial distribution, each trial has only two possible outcomes: success or failure. These are fundamental concepts to understand while studying binomial probability.

A **success** has been defined in this exercise as a questionnaire being returned, with probability \(p=0.7\).

Conversely, a **failure** occurs when the questionnaire is not returned, which would have a probability of \(1-p = 0.3\).

These terms must be clearly defined because they directly influence how calculations are performed using the binomial probability formula. Success and failure probabilities give insight into chance and help forecast the likely outcome of repeating experiments under the same conditions.

Probability Mass Function

A Probability Mass Function (PMF) specifies the probability of a discrete random variable taking on a specific value. For binomially distributed random variables, the PMF uses the binomial probability formula to determine the probabilities across the range of possible successes.

The PMF evaluates the probability for each possible success count between 0 and \(n\) in this distribution. Each probability found from the PMF reflects on the likelihood of achieving exactly that number of successes.

In this exercise, we use the PMF to compute the probability for specific counts of questionnaires returned. For instance, \(P(X=10)\), \(P(X \geq 12)\), and \(P(X \leq 10)\) are assessed using the individual probabilities that come from the PMF.