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 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?

Assume that a student is chosen at random from a class. Determine whether the events $A$ and $B$ are independent, mutually exclusive, or neither.

A: The student is a freshman.

B: The student is a sophomore

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 $P(V \cup W)=0.17.$ a. Find the probability that the computer contains both a virus and a worm. b. Find the probability that the computer contains neither a virus nor a worm. c. Find the probability that the computer contains a virus but not a worm.

For each definite integral: a. Evaluate it using the table of integrals on the inside back cover. (Leave answers in exact form.) b. Use a graphing calculator to verify your answer to part (a). $\int_{3}^{5} \frac{\sqrt{25-x^{2}}}{x} d x$.

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)

Combine and simplify.

$$\frac{7 y}{12}+\frac{9 y}{12}$$

In each of the following data sets, tell whether the outlier seems certain to be due to an error, or whether it could conceivably be correct. a. The length of a rod is measured five times. The readings in centimeters are 48.5, 47.2, 4.91, 49.5, 46.3. b. The prices of five cars on a dealer’s lot are $25,000,$30,000, $42,000,$110,000, $31,000.

One card is drawn from a standard deck of playing cards. Let E1 be the event that the card is a spade, and let E2 be the event the card is a heart.

What is the probability of E1 or E2?

A car dealer sold 750 automobiles last year. The following table categorizes the cars sold by size and color and presents the number of cars in each category. A car is to be chosen at random from the 750 for which the owner will win a lifetime of free oil changes.

$$\begin{matrix} & \text{Color}\\ \text{Size} & \text{White} & \text{Black} & \text{Red} & \text{Grey}\\ \text{Small} & \text{102} & \text{71} & \text{33} & \text{134}\\ \text{Midsize} & \text{86} & \text{63} & \text{36} & \text{105}\\ \text{Large} & \text{26} & \text{32} & \text{22} & \text{40}\\ \end{matrix}$$

a. If the car is small, what is the probability that it is black? b. If the car is white, what is the probability that it is midsize? c. If the car is large, what is the probability that it is red? d. If the car is red, what is the probability that it is large? e. If the car is not small, what is the probability that it is not grey?

Based on data from the Statistical Abstract of the United States, 112 th Edition, only about 14% of senior citizens (65 years old or older) get the flu each year. However, about 24% of the people under 65 years old get the flu each year. In the general population, there are 12.5% senior citizens (65 years old or older). What is the probability that a person selected at random from the general population is a person under age 65 who will get the flu this year?

In the process of producing engine valves, the valves are subjected to a first grind. Valves whose thicknesses are within the specification are ready for installation. Those valves whose thicknesses are above the specification are reground, while those whose thicknesses are below the specification are scrapped. Assume that after the first grind, 70% of the valves meet the specification, 20% are reground, and 10% are scrapped. Furthermore, assume that of those valves that are reground, 90% meet the specification, and 10% are scrapped. a. Find the probability that a valve is ground only once. b. Given that a valve is not reground, what is the probability that it is scrapped? c. Find the probability that a valve is scrapped. d. Given that a valve is scrapped, what is the probability that it was ground twice? e. Find the probability that the valve meets the specification (after either the first or second grind). f. Given that a valve meets the specification (after either the first or second grind), what is the probability that it was ground twice? g. Given that a valve meets the specification, what is the probability that it was ground only once?

Of all failures of a certain type of computer hard drive, it is determined that in 20% of them only the sector containing the file allocation table is damaged, in 70% of them only nonessential sectors are damaged, and in 10% of the cases both the allocation sector and one or more nonessential sectors are damaged. A failed drive is selected at random and examined. a. What is the probability that the allocation sector is damaged? b. What is the probability that a nonessential sector is damaged? c. If the drive is found to have a damaged allocation sector, what is the probability that some nonessential sectors are damaged as well? d. If the drive is found to have a damaged nonessential sector, what is the probability that the allocation sector is damaged as well? e. If the drive is found to have a damaged allocation sector, what is the probability that no nonessential sectors are damaged? f. If the drive is found to have a damaged nonessential sector, what is the probability that the allocation sector is not damaged?

Sarah and Thomas are going bowling. The probability that Sarah scores more than 175 is 0.4, and the probability that Thomas scores more than 175 is 0.2. Their scores are independent. a. Find the probability that both score more than 175. b. Given that Thomas scores more than 175, the probability that Sarah scores higher than Thomas is 0.3. Find the probability that Thomas scores more than 175 and Sarah scores higher than Thomas.