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 $P(7)$.

Let $A$ and $B$ be events with $P (A) = 0.4, P (B) = 0.7,$ and $P (B | A) = 0.3$. Find $P (A\ \text{and}\ B)$.

Simplify: 0.24 × 0.15 × 0.05

Determine whether events A and B are mutually exclusive.

A: Sophie is a member of the debate team; B: Sophie is the president of the theater club

Let $E$ be the event that a new car requires engine work under warranty and let T be the event that the car requires transmission work under warranty. Suppose that $P (E) = 0.10$, $P (T) = 0.02$, and $P (E\ \text{and}\ T) = 0.01$.

$\textbf{a.}\hspace{10pt}$ Find the probability that the car needs work on either the engine, the transmission, or both.

$\textbf{b.}\hspace{10pt}$ Find the probability that the car needs no work on the engine.

Suppose I bet on red at roulette and you bet on black, both bets on the same spin of the wheel.

a) What is the probability that we both lose?

b) What is the probability that at least one of us wins?

c) What is the probability that at least one of us loses?

A Nevada roulette wheel has $38$ pockets. Eighteen of them are red, eighteen are black, and two are green. Each time the wheel is spun, a ball lands in one of the pockets, and each pocket is equally likely.

$\textbf{a.}\hspace{10pt}$ What is the probability that the ball lands in a green pocket?

$\textbf{b.}\hspace{10pt}$ If you bet on green on every spin of the wheel, you will lose more than $90\%$ of the time in the long run. Explain why this is so.

A Nevada roulette wheel has $38$ pockets. Eighteen of them are red, eighteen are black, and two are green. Each time the wheel is spun, a ball lands in one of the pockets, and each pocket is equally likely.

$\textbf{a.}\hspace{10pt}$ What is the probability that the ball lands in a red pocket?

$\textbf{b.}\hspace{10pt}$ If you bet on red on every spin of the wheel, you will lose more than half the time in the long run. Explain why this is so.

An investor is considering a $\$10,000$ investment in a start-up company. She estimates that she has probability $0.25$ of a $\$20,000$ loss, probability $0.20$ of a $\$10,000$ profit, probability $0.15$ of a $\$50,000$ profit, and probability $0.40$ of breaking even (a profit of $\$0$). What is the expected value of the profit? Would you advise the investor to make the investment?

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 $P (2)$.

Suppose that A and B are two events. a. What does it mean for event B to be independent of event A? b. If event A and event B are independent, how can their joint probability be obtained from their marginal probabilities?

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 lot contains 15 castings from a local supplier and 25 castings from a supplier in the next state. Two castings are selected randomly, without replacement, from the lot of 40. Let A be the event that the first casting selected is from the local supplier, and let B denote the event that the second casting is selected from the local supplier. Determine: a. P(A) b. P(B \ A) c. P(A ∩ B) d. P(A ∪ B)