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.

In a semiconductor manufacturing process, three wafers from a lot are tested. Each wafer is classified as pass or fail. Assume that the probability that a wafer passes the test is 0.8 and that wafers are independent. Determine the probability mass function of the number of wafers from a lot that pass the test.

A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop after the end of a run. Suppose that the main chute deploys with probability 0.99, and that if the main fails to deploy, the backup deploys with probability 0.98. a. What is the probability that one of the two parachutes deploys? b. What is the probability that the backup parachutes deploys?

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

$$\begin{array}{lrrrr} \hline & & & {\text { Region }} \\ \hline \text { Population of City } & \text { NE } & \text { SE } & \text { SW } & \text { NW } \\ \hline \text { Under } 50,000 & 30 & 35 & 15 & 5 \\ 50,000-500,000 & 60 & 90 & 70 & 30 \\ \text { Over } 500,000 & 150 & 25 & 30 & 60 \\ \hline \end{array}$$

a. If the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast? b. If the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 50,000? c. If 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? d. If the restaurant is located in a city with a population of 500,000 or less, what is the probability that is in the Southwest? e. If 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?

A penny and a nickel are tossed. The penny has probability 0.4 of coming up heads, and the nickel has probability 0.6 of coming up heads. Let X = 1 if the penny comes up heads, and let X = 0 if the penny comes up tails. Let Y = 1 if the nickel comes up heads, and let Y = 0 if the nickel comes up tails. a. Find the probability mass function of X. b. Find the probability mass function of Y. c. Is it reasonable to assume that X and Y are independent? Why? d. Find the joint probability mass function of X and Y.

Suppose that start-up companies in the area of biotechnology have probability 0.2 of becoming profitable, and that those in the area of information technology have probability 0.15 of becoming profitable. A venture capitalist invests in one firm of each type. Assume the companies function independently. a. What is the probability that both companies become profitable? b. What is the probability that neither company becomes profitable? c. What is the probability that at least one of the two companies become profitable?

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?

An electronic message consists of a string of bits (0s and 1s). The message must pass through two relays before being received. At each relay the probability is 0.1 that the bit will be reversed before being relayed (i.e., a 1 will be changed to a 0, or a 0 to a 1). Find the probability that the value of a bit received at its final destination is the same as the value of the bit that was sent.

A quality-control engineer samples 100 items manufactured by a certain process, and finds that 15 of them are defective. True or false: a. The probability that an item produced by this process is defective is 0.15. b. The probability that an item produced by this process is defective is likely to be close to 0.15, but not exactly equal to 0.15.

The following table displays the $100$ senators of the $115$th U.S. Congress on January $3, 2017$, classified by political party affiliation and gender.

$$\begin{array}{lccc} \hline & \textbf{Male} & \textbf{Female} & \textbf{Total}\\ \hline \textbf{Democrat} & 30 & 16 & 46\\ \textbf{Republican} & 47 & 5 & 52\\ \textbf{Independent} & 2 & 0 & 2\\ \hline \textbf{Total} & 79 & 21 & 100\\ \end{array}$$

A senator is selected at random from this group. Compute the following probabilities.

$\textbf{a.}\hspace{10pt}$ The senator is a male Republican.

$\textbf{b.}\hspace{10pt}$ The senator is a Democrat or a female.

$\textbf{c.}\hspace{10pt}$ The senator is a Republican.

$\textbf{d.}\hspace{10pt}$ The senator is not a Republican.

$\textbf{e.}\hspace{10pt}$ The senator is a Democrat.

$\textbf{f.}\hspace{10pt}$ The senator is an Independent.

$\textbf{g.}\hspace{10pt}$ The senator is a Democrat or an Independent.

Given that:

• Probability of a type $1$ defect is $0,2$.
• Probability of type $2$ defect is $0.4$.
• Probability of type $3$ defect is $0.2$.
• Probability of both type $1$ and $2$ defect is $0.3$
• Probability of all three types of defect is $0.1$

What is the probability that the system has both type $1$ and type $2$ defects but not a type $3$ defect?

A system has n independent units, each of which fails with probability p. The system fails only if k or more of the units fail. What is the probability that the system fails?

Assume that an electronic system contains 'n' components that function independently of each other and that the probability that component 'i' will function properly is $p_i$(i=1,.., n). It is said that the components are connected in series if a necessary and sufficient condition for the system to function properly is that all $n$ components function properly. It is said that the components are connected in parallel if a necessary and sufficient condition for the system to function properly is that at least one of the $n$ components functions properly. The probability that the system will function properly is called the reliability of the system. Determine the reliability of the system,

a) assuming that the components are connected in series,

b) assuming that the components are connected in parallel.

A geneticist is studying two genes. Each gene can be either dominant or recessive. A sample of 100 individuals is categorized as follows.

$$\begin{matrix} & \text{Gene 2}\\ \text{Gene 1} & \text{Dominant} & \text{Recessive}\\ \text{Dominant} & \text{56} & \text{24}\\ \text{Recessive} & \text{14} & \text{6}\\ \end{matrix}$$

a. What is the probability that a randomly sampled individual, gene 1 is dominant? b. What is the probability that a randomly sampled individual, gene 2 is dominant? c. Given that gene 1 is dominant, what is the probability that gene 2 is dominant? d. These genes are said to be in linkage equilibrium if the event that gene 1 is dominant is independent of the event that gene 2 is dominant. Are these genes in linkage equilibrium?