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?

The data shown represent the number of outdoor drive-in movies in the United States for a 14-year period. Check for normality.

$$\begin{array}{rrrrrrrr} 2084 & 1497 & 1014 & 910 & 899 & 870 & 837 & 859 \\ 848 & 826 & 815 & 750 & 637 & 737 & & \end{array}$$

Let A and B be independent events with P(A) = 0.7 and P(B) = 0.2. Compute (a) P(A ∩ B), (b) P(A ∪ B), and (c) P(A' ∪ B').

A student council consists of three freshmen, four sophomores, four juniors, and five seniors. How many committees of eight members of the council contain at least one member from each class?

A student would like to demonstrate destructive interference using two sound sources. Explain how the student could set up this demonstration and what restrictions they would need to place upon their sources. Be sure to consider both the layout of space and the sounds created in your explanation.

An unfair coin has probability $0.4$ of landing heads. The coin is tossed four times. What is the probability that it lands heads at least once?

A group of friends have reserved a tennis court. They find that there are ten different ways in which two of them can play a singles game on this court. How many friends are in this group?

If State's football team has a 10% chance of winning Saturday's game, a 30% chance of winning two weeks from now, and a 65% chance of losing both games, what are their chances of winning exactly once?

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.

The Student Council at a certain school has $10$ members. Four members will form an executive committee consisting of a president, a vice president, a secretary, and a treasurer.

$\textbf{a.}\hspace{10pt}$ In how many ways can these four positions be filled?

$\textbf{b.}\hspace{10pt}$ In how many ways can four people be chosen for the executive committee if it does not matter who gets which position?

$\textbf{c.}\hspace{10pt}$ Four of the people on Student Council are Zachary, Yolanda, Xavier, and Walter. What is the probability that Zachary is president, Yolanda is vice president, Xavier is secretary, and Walter is treasurer?

$\textbf{d.}\hspace{10pt}$ What is the probability that Zachary, Yolanda, Xavier, and Walter are the four committee members?

The number of times that screen flicker appears in video recorded on a certain type of digital video recorder (DVR) has a Poisson distribution with a rate of $0.2$ per minute.

$\textbf{a.}$ What is the probability that there are no screen flickers in $8$ minutes of video?

$\textbf{b.}$ What is the probability that there are at least $2$ screen flickers in $11$ minutes of video?

The Centers for Disease Control and Prevention reports that $25\%$ of baby boys $6-8$ months old in the United States weigh more than $20$ pounds. A sample of $16$ babies is studied.

$\textbf{a.}\hspace{10pt}$ What is the probability that exactly $5$ of them weigh more than $20$ pounds?

$\textbf{b.}\hspace{10pt}$ What is the probability that more than $6$ weigh more than $20$ pounds?

$\textbf{c.}\hspace{10pt}$ What is the probability that fewer than $3$ weigh more than $20$ pounds?

$\textbf{d.}\hspace{10pt}$ Would it be unusual if more than $8$ of them weigh more than $20$ pounds?

$\textbf{e.}\hspace{10pt}$ What is the mean number who weigh more than $20$ pounds in a sample of $16$ babies aged $6-8$ months?

$\textbf{f.}\hspace{10pt}$ What is the standard deviation of the number who weigh more than $20$ pounds in a sample of $16$ babies aged $6-8$ months?

Foreign Adoptions The following foreign adoptions (in the United States) occurred during these particular years.

$$\begin{array}{lrr} & 2014 & \mathbf{2 0 1 3} \\ \hline \textbf{China} & 2040 & 2306 \\ \textbf{Ethiopia} & 716 & 993 \\ \textbf{Ukraine} & 521 & 428 \\ \end{array}$$

Choose one adoption at random from this chart.
Choose two adoptions at random; what is the probability that they were both from China?

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?