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Chapter 4: Problem 99

Suppose that \(20 \%\) of all adults in a small town live alone, and \(8 \%\) of the adults live alone and have at least one pet. What is the probability that a randomly selected adult from this town has at least one pet given that this adult lives alone?

Short Answer

Expert verified
The probability that a randomly selected adult from this town has at least one pet given that this adult lives alone is \(40\% \) or \(0.40\)

Step by step solution

01

Problem Analysis

First, identify the two events in question. Let A = event that an adult lives alone. This has a probability of \(0.20\) or \(20\% \). Let B = event that an adult has a pet and lives alone. This has a probability of \(0.08\) or \(8\% \).
02

Use the formula for conditional probability

The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \) . Given that an adult lives alone, we want to find the likelihood that they also have at least one pet, i.e., \( P(B|A) \).
03

Substitute values into the formula

Substituting the values into the formula, we get \( P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.08}{0.20} \).
04

Simplify the fraction

Simplifying the fraction gives us \( P(B|A) = 0.40 \) or \( 40\% \).

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Most popular questions from this chapter

Five hundred employees were selected from a city's large private companies, and they were asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$\begin{array}{lcc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \end{array}$$ a. If one employee is selected at random from these 500 employees, find the probability that this employee i. is a woman ii. has retirement benefits iii. has retirement benefits given the employee is a man iv. is a woman given that she does not have retirement benefits b. Are the events "man" and "yes" mutually exclusive? What about the events "yes" and "no?" Why or why not? c. Are the events "woman" and "yes" independent? Why or why not?

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Five hundred employees were selected from a city's large private companies and asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$\begin{array}{llc} \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array}$$ a. Suppose one employee is selected at random from these 500 employees. Find the following probabilities. i. Probability of the intersection of events "woman" and "yes" ii. Probability of the intersection of events "no" and "man" b. Mention what other joint probabilities you can calculate for this table and then find them. You may draw a tree diagram to find these probabilities.

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