EXERCISES Section 3.5 (68--78)

68

Eighteen individuals are scheduled to take a driving test at a particular DMV office on a certain day, eight of whom will be taking the test for the first time. Suppose that six of these individuals are randomly assigned to a particular examiner, and let $X$ be the number among the six who are taking the test for the first time.

a. What kind of a distribution does $X$ have (name and values of all parameters)?

b. Compute $P\left(X=2\right),P\left(X\le 2\right)$ , and $P\left(X\ge 2\right)$ .

c. Calculate the mean value and standard deviation of $X$ .

69

Each of 12 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 7 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let $X$ be the number among the first 6 examined that have a defective compressor.

a. Calculate $P\left(X=4\right)$ and $P\left(X\le 4\right)$

b. Determine the probability that $X$ exceeds its mean value by more than 1 standard deviation.

c. Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If $X$ is the number among 15 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) $P\left(X\le 5\right)$ than to use the hypergeometric pmf.

70

An instructor who taught two sections of engineering statistics last term, the first with 20 students and the second with 30 , decided to assign a term project. After all projects had been turned in, the instructor randomly ordered them before grading. Consider the first 15 graded projects.

a. What is the probability that exactly 10 of these are from the second section?

b. What is the probability that at least 10 of these are from the second section?

c. What is the probability that at least 10 of these are from the same section?

d. What are the mean value and standard deviation of the number among these 15 that are from the second section?

e. What are the mean value and standard deviation of the number of projects not among these first 15 that are from the second section?

71

A geologist has collected 10 specimens of basaltic rock and 10 specimens of granite. The geologist instructs a laboratory assistant to randomly select 15 of the specimens for analysis.

a. What is the pmf of the number of granite specimens selected for analysis?

b. What is the probability that all specimens of one of the two types of rock are selected for analysis?

c. What is the probability that the number of granite specimens selected for analysis is within 1 standard deviation of its mean value?

72

A personnel director interviewing 11 senior engineers for four job openings has scheduled six interviews for the first day and five for the second day of interviewing. Assume that the candidates are interviewed in random order.

a. What is the probability that $x$ of the top four candidates are interviewed on the first day?

b. How many of the top four candidates can be expected to be interviewed on the first day?

73

Twenty pairs of individuals playing in a bridge tournament have been seeded $1,\dots ,20$ . In the first part of the tournament, the 20 are randomly divided into 10 east-west pairs and 10 north-south pairs.

a. What is the probability that $x$ of the top 10 pairs end up playing east-west?

b. What is the probability that all of the top five pairs end up playing the same direction?

c. If there are $2n$ pairs, what is the pmf of $X=$ the number among the top $n$ pairs who end up playing east-west? What are $E\left(X\right)$ and $V\left(X\right)$ ?

74

A second-stage smog alert has been called in a certain area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations.

a. If 15 of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation?

b. If there are 500 firms in the area, of which 150 are in violation, approximate the pmf of part (a) by a simpler pmf.

c. For $X=$ the number among the 10 visited that are in violation, compute $E\left(X\right)$ and $V\left(X\right)$ both for the exact pmf and the approximating pmf in part (b).

75

The probability that a randomly selected box of a certain type of cereal has a particular prize is .2. Suppose you purchase box after box until you have obtained two of these prizes.

a. What is the probability that you purchase $x$ boxes that do not have the desired prize?

b. What is the probability that you purchase four boxes?

c. What is the probability that you purchase at most four boxes?

d. How many boxes without the desired prize do you expect to purchase? How many boxes do you expect to purchase?

76

A family decides to have children until it has three children of the same gender. Assuming $P\left(B\right)=P\left(G\right)=.5$ , what is the pmf of $X=$ the number of children in the family?

77

Three brothers and their wives decide to have children until each family has two female children. What is the pmf of $X=$ the total number of male children born to the brothers? What is $E\left(X\right)$ , and how does it compare to the expected number of male children born to each brother?

78

According to the article “Characterizing the Severity and Risk of Drought in the Poudre River, Colorado” (J. of Water Res. Planning and Mgmnt., 2005: 383-393), the drought length $Y$ is the number of consecutive time intervals in which the water supply remains below a critical value $y_0$ (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). The cited paper proposes a geometric distribution with $p=.409$ for this random variable.

a. What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals?

b. What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?