## Related questions with answers

A fast-food restaurant chain has $600$ outlets in the United States. The following table categorizes them by city population size and location, and presents the number of restaurants in each category. A restaurant is to be chosen at random from the $600$ to test market a new menu.

$\begin{array}{lr} \hline \text{Population}\hspace{120pt} & \text{Region}\hspace{30pt} \end{array}\\ \begin{array}{lcccc} \text{of city} & \hspace{60pt}\text{NE} & \text{SE} & \text{SW} & \text{NW}\\ \hline \textbf{Under 50,000} & \hspace{60pt}30 & 35 & 15 & 5\\ \textbf{50,000 - 500,000} & \hspace{60pt}60 & 90 & 70 & 30\\ \textbf{Over 50,000} & \hspace{60pt}150 & 25 & 30 & 60\\ \hline \end{array}$

$\textbf{a.}\hspace{10pt}$ Given that the restaurant is located in a city with a population over $500,000$, what is the probability that it is in the Northeast?

$\textbf{b.}\hspace{10pt}$ Given that the restaurant is located in the Southeast, what is the probability that it is in a city with a population under $50,000$?

$\textbf{c.}\hspace{10pt}$ Given that the restaurant is located in the Southwest, what is the probability that it is in a city with a population of $500,000$ or less?

$\textbf{d.}\hspace{10pt}$ Given that the restaurant is located in a city with a population of $500,000$ or less, what is the probability that it is in the Southwest?

$\textbf{e.}\hspace{10pt}$ Given that the restaurant is located in the South (either SE or SW), what is the probability that it is in a city with a population of $50,000$ or more?

Solution

Verified**(a)**

To determine the probability that a restaurant is in the Northeast, given that the restaurant is located in a city with over $500,000$ population we will use the general method for computing conditional probabilities.

$P(B|A)=\frac{P(A \ \text{and} \ B)}{P(A)}$

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