The following table presents the number of reports of graffiti in each of New York’s five boroughs over a one-year period. These reports were classified as being open, closed, or pending.

$$\begin{array}{lrrrr} \hline & \text{Open} & \text{Closed} & \text{Pending} & \\ \text{Borough} & \text{Reports} & \text{Reports} & \text{Reports} & \text{Total}\\ \hline \text{Bronx} & 1,121 & 1,622 & 80 & 2,823\\ \text{Brooklyn} & 1,170 & 2,706 & 48 & 3,924\\ \text{Manhattan} & 744 & 3,380 & 25 & 4,149\\ \text{Queens} & 1,353 & 2,043 & 25 & 3,421\\ \text{Staten Island} & 83 & 118 & 0 & 201\\ \hline \text{Total} & 4,471 & 9,869 & 178 & 14,518\\ \end{array}$$

A graffiti report is selected at random. Compute the following probabilities.

$\textbf{a.}\hspace{10pt}$ The report is open and comes from Brooklyn.

$\textbf{b.}\hspace{10pt}$ The report is closed or comes from Queens.

$\textbf{c.}\hspace{10pt}$ The report comes from Manhattan.

$\textbf{d.}\hspace{10pt}$ The report does not come from Manhattan.

$\textbf{e.}\hspace{10pt}$ The report is pending.

$\textbf{f.}\hspace{10pt}$ The report is from the Bronx or Staten Island.

A fast-food chain estimates that if it opens a restaurant in a shopping center, the probability that the restaurant is successful is 0.72. A successful restaurant earns an annual profit of $120,000; a restaurant that is not successful loses$36,000. What is the expected gain to the chain if it opens a restaurant in a shopping center?

A quiz consists of three true–false questions and two multiple-choice questions with five choices each. How many different sets of answers are there?

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?

Assume that a fair die is rolled. The sample space is $\{1,2,3,4,5,6\}$, and all the outcomes are equally likely.

Find $P(7)$.

How many ways can a $3$-person subcommittee be selected from a committee of $7$ people? How many ways can a president, vice-president, and secretary be chosen from a committee of $7$ people?

37. In how many ways can a committee of four men and five women be formed from a group of seven men and seven women?
38. How many different committees can be formed from 5 professors and 15 students if each committee is made up of 2 professors and 10 students?

A population of $600$ semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification, with the results shown in the following table. A wafer is chosen at random from the population.

$$\begin{array}{crr} \hline \textbf{Lot} & \textbf{Conforming} & \textbf{Nonconforming}\\ \hline \textbf{A} & 88\hspace{20pt} & 12\hspace{30pt}\\ \textbf{B} & 165\hspace{20pt} & 35\hspace{30pt}\\ \textbf{C} & 260\hspace{20pt} & 40\hspace{30pt}\\ \hline \end{array}$$

$\textbf{a.}\hspace{10pt}$ What is the probability that a wafer is from Lot A?

$\textbf{b.}\hspace{10pt}$ What is the probability that a wafer is conforming?

$\textbf{c.}\hspace{10pt}$ What is the probability that a wafer is from Lot A and is conforming?

$\textbf{d.}\hspace{10pt}$ Given that the wafer is from Lot A, what is the probability that it is conforming?

$\textbf{e.}\hspace{10pt}$ Given that the wafer is conforming, what is the probability that it is from Lot A?

$\textbf{f.}\hspace{10pt}$ Let $E_1$ be the event that the wafer comes from Lot A, and let $E_2$ be the event that the wafer is conforming. Are $E_1$ and $E_2$ independent? Explain.

A population of 600 semiconductor wafers contains wafers from three lots. The wafers are categorized by lot and by whether they conform to a thickness specification. The following table presents the number of wafers in each category. A wafer is chosen at random from the population.

$$\begin{matrix} \text{Lot} & \text{Conforming} & \text{Nonconforming}\\ \text{A} & \text{88} & \text{12}\\ \text{B} & \text{165} & \text{35}\\ \text{C} & \text{260} & \text{40}\\ \end{matrix}$$

a. If the wafer is from Lot A, what is the probability that it is conforming? b. If the wafer is conforming, what is the probability that it is from Lot A? c. If the wafer is conforming, what is the probability that it is not from Lot C? d. If the wafer is not from Lot C, what is the probability that it is conforming?

A section of an exam contains four True-False questions. A completed exam paper is selected at random, and the four answers are recorded. a. List all 16 outcomes in the sample space. b. Assuming the outcomes to be equally likely, find the probability that all the answers are the same. c. Assuming the outcomes to be equally likely, find the probability that exactly one of the four answers is “True.” d. Assuming the outcomes to be equally likely, find the probability that at most one of the four answers is “True.”

A company audit showed that of $875$ bills that were sent out, $623$ were paid on time, $155$ were paid up to $30$ days late, $78$ were paid between $31$ and $90$ days late, and $19$ were paid after $90$ days. One bill is selected at random.

$\textbf{a.}\hspace{10pt}$ What is the probability that the bill was paid on time?

$\textbf{b.}\hspace{10pt}$ What is the probability that the bill was paid late?

Let $V$ be the event that a computer contains a virus, and let W be the event that a computer contains a worm. Suppose $P (V) = 0.15$, $P (W) = 0.05$, and $P (V\ \text{and}\ W) = 0.03$.

$\textbf{a.}\hspace{10pt}$ Find the probability that the computer contains either a virus or a worm or both.

$\textbf{b.}\hspace{10pt}$ Find the probability that the computer does not contain a virus.

Assume that a fair die is rolled. The sample space is $\{1,2,3,4,5,6\}$, and all the outcomes are equally likely.

Find $P (2)$.

A utility company sends surveys to $200$ of its customers in such a way that $100$ surveys are sent to customers who pay their bills on time, $50$ surveys are sent to customers whose bills are less than $30$ days late, and $50$ surveys are sent to customers whose bills are more than $30$ days late. Which type of sample does this represent?