# EBook Problems Prob Rules

## EBook Problems Set - Rules for Computing Probabilities

### Problem 1

A professor who teaches 500 students in an introductory psychology course reports that 250 of the students have taken at least one introductory statistics course, and the other 250 have not taken any statistics courses. 200 of the students were freshmen, and the other 300 students were not freshmen. Exactly 50 of the students were freshmen who had taken at least one introductory statistics course. If you select one of these psychology students at random, what is the probability that the student is not a freshman and has never taken a statistics course?

(a) 30%
(b) 40%
(c) 50%
(d) 60%
(e) 20%

### Problem 2

A box contains 30 pens, where 5 are red, 14 are black, and 11 are blue. If you pick three pens from the box at random without replacement, what is the probability that these three pens will all be black?

(a) 14/30 + 14/30 + 14/30
(b) 14/30 + 13/29 + 12/28
(c) 14/30 x 13/29 x 12/28
(d) 1 - (14/30 x 13/29 x 12/28)

### Problem 3

When three fair dice are simultaneously thrown, which of these three results is least likely to be obtained?

(a) All three results are equally unlikely.
(b) Two fives and a 3 in any order.
(c) A 5, a 3 and a 6 in any order.
(d) Three 5's.

### Problem 4

Records show that in an introductory chemistry course in a college, 20% of the students get an A, 30% get a B, 40% get a C, and 10% get a D. If you pick three students at random, what is the probability that all three will get an A?

(a) 0.8*0.8*0.8
(b) 0.2*0.2*0.2
(c) 200*0.2*0.2*0.2
(d) 0.2*3

### Problem 5

A newly born child is equally likely to be a boy or a girl. What is the probability that in a family of three children there are less than 3 boys?

(a) 0.125
(b) 0.75
(c) 0.875
(d) 0.5

### Problem 6

A professor who teaches 300 students in an introductory psychology course reports that 135 of the students have taken exactly one introductory statistics course, 60 have taken two or more introductory statistics courses, and the other 105 have not taken any statistics courses. If you select one of these psychology students at random, what is the probability that the student has taken at least one statistics class?

(a) 0.20
(b) 0.45
(c) 0.65
(d) 0.35

### Problem 7

Two fair coins are flipped. The probability that both are heads is:

(b) Exactly 25%
(c) Exactly 12.5%
(d) Exactly 50%
(e) Exactly 75%

### Problem 8

Two fair coins are flipped. The probability that the second coin is a head, given that the first was a head, is:

(a) Exactly 50%
(b) Exactly 25%
(c) Exactly 75%
(d) Exactly 12.5%

### Problem 9

In a carnival game, a person can win a prize by guessing which one of 5 identical boxes contains the prize. After each guess, if the prize has been won, a new prize is randomly placed in one of the 5 boxes. If a person makes 4 guesses, what is the probability that the person wins a prize exactly twice?

(a) (0.2)^2/(0.8)^2
(b) 2(0.2)^2*(0.8)^2
(c) 6(0.2)^2*(0.8)^2
(d) (0.2)^2*(0.8)^2
(e) 2!/5!

### Problem 10

Suppose that you read that 88% of all college students have working cell phones. If you take a random sample of 20 college students, what are the chances that at least 18 of the 20 will have working cell phones?

(a) 0.88*20
(b) 20!/(18!*2!) (0.88)^18 (0.12)^2
(c) 190*(0.88)^18 (0.12)^2 + 20*(0.88)^19(0.12) + (0.88)^20
(d) 1 - 190*(0.88)^18 (0.12)^2 + 20*(0.88)^19(0.12) + (0.88)^20
(e) None of these values

### Problem 11

Express Checkout Researchers report that a quarter of the customers who shop for groceries between 5 P.M. and 7 P.M. use express checkout for 10 items or less. Consider 5 randomly selected customers, and let x represent those who use express checkout. Determine the probability that exactly 2 of the 5 customers use express checkout?

(a) (0.25)^2
(b) 10*(0.25)^2*(0.75)^3
(c) 10*(0.25)^2(0.75)^3 + 5*(0.25)^3(0.75)^2 + 5*(0.25)^4(0.75) + (0.25)^5
(d) Cannot be determined without the data.

### Problem 12

Imagine that you are taking a multiple-choice test in a language which is completely unfamiliar to you. Suppose the test has five questions with four choices for each answer and only one of the four choices is correct. If you answer each question independently and at random, what is the probability that you will answer all five questions incorrectly?

(a) 3/4*3/4*3/4*3/4*3/4
(b) 1- (3/4*3/4*3/4*3/4*3/4)
(c) 3/4 + 3/4 + 3/4 + 3/4 + 3/4
(d) 1- (1/4*1/4*1/4*1/4*1/4)

### Problem 13

In a recent issue of Science magazine, we read about a new computer-based test for ovarian cancer, "clinical proteomics" that exams a blood sample for the presence of certain patterns of proteins. Ovarian cancer, though dangerous, is very rare, affecting only one in 5000 women. The test is highly sensitive, able to correctly detect the present of ovarian cancer in 99.97% of women who have the disease. However, it is unlikely to be used as a screening test for the general public, because the test gives a false positive 5% of the time. Draw a tree diagram and determine the probability that a woman tests positive using this method.

(a) 0.0998*0.05
(b) 0.0002*0.9997 + 0.9998*0.05
(c) (0.9998*0.95)/(0.0002*0.0003 +0.998*0.95)
(d) (0.0002*0.9997)/(0.0002*0.9997 + 0.9998*0.05)
(e) 0.0002*0.9997

### Problem 14

A quality control engineer claims that the number of defective chips manufactured during runs of 100 of these products is independent of the number of runs and that the proportion of defects is about 4 per 100. What is the probability that there will be no defects in the next run of 100 chips?

(a) 0.009
(b) None of these values
(c) 0.051
(d) 0.034
(e) 0.017

### Problem 15

In a university with 20,000 students, 20% are engineering students, 40% are in the sciences, 30% are in the social sciences, and the rest have other majors. The counselors in the registrar's office want to survey the opinions of students on the issue of posting grades on-line and they seek opinions from students of various majors. They conduct a survey by randomly selecting students. Among the first three student selected, what is the probability that two of the three major in social sciences and one has a major other than social science?

(a) 0.600
(b) 0.189
(c) 0.090
(d) 0.063

### Problem 16

Suppose that at a local community college, 90% of the students take English. Of the students who do not take English, 80% take art classes. Of the students who take English, 50% take art. What is the probability that a student at this college takes art?

(a) 0.53
(b) 0.13
(c) 0.80
(d) 0.50
(e) 0.45

### Problem 17

A fast food restuarant reports that: 60% of the customers order white meat, 20% of those who order white meat also order a soft drink, 60% of those who do not order white meat order a soft drink. If an individual orders a soft drink, what is the probability that he ordered white meat?

(a) 0.33
(b) 0.36
(c) 0.24
(d) 0.12

### Problem 18

Imagine that you take a three question "true/false" quiz for which you are completely unprepared. You have to guess the answer for each question. What is the probability of answering at least one of the three questions correctly?

(a) 1/8
(b) 7/8
(c) 3/8
(d) 5/8
(e) 4/8

### Problem 19

Every five years the Conference Board of Mathematical Sciences surveys college math departments. In a recent report, 51% of all undergraduates taking calculus were in classes using graphing calculators and 31% were in classes using computer assignments. Suppose that 16% of these students use both calculator and computer. What proportion of undergraduates taking calculus use no technology?

(a) 0.44
(b) 0.82
(c) 0.34
(d) 0.66
(e) 0.16

### Problem 20

Let's look at the residents of Los Angeles (Angelinos). Suppose 70% of Angelinos bike (B), 30% like ice cream (I), and 40% regularly see new movie releases (M). In addition, assume that 15% B and I, 40% J and M, 20% I and M, and 5% like all 3 (B and I and M). Find the probability that a randomly chosen Angelino likes at least one of these three activities.

$$P(B\cup I \cup M)=P(B)+P(I)+P(M)-P(B\cap I)-P(I\cap M)-P(B\cap M)+P(B\cap I\cap M)=1.4-0.75+0.05=0.7$$
(a) 1.4
(b) 0.75
(c) 0.8
(d) 0.7
(e) 0.05