MAT 104 Pace University Mathematics Exam Questions

Pace UniversityFall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
Chapter 5
1. In a survey of 75 college students, it was found that of the three weekly news magazines
Time, Newsweek, and U.S. News & World Report:
23 read Time
18 read Newsweek
14 read U.S. News & World Report
10 read Time and Newsweek
9 read Time and U.S. News & World Report
8 read Newsweek and U.S. News & World Report
5 read all three
a. How many people read none of these three magazines?
b. How many read Time alone?
c. How many read Newsweek alone?
d. How many read neither Time nor Newsweek?
2. Mrs. Bollo’s second grade class of thirty students conducted a pet ownership survey.
Results of the survey indicate that 8 students own a cat, 15 students own a dog, and 5
students own both a cat and a dog. How many of the students surveyed own a cat or a
dog?
1
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
3. Simplify the expression 𝑆 ∪ (𝑆 ′ ∩ 𝑇 ′ )′ using De Morgan’s Laws.
4. Are (𝑅 ∪ 𝑆 ′ ) ∩ 𝑇 and (R′ ∪ 𝑆) ∪ 𝑇′ equal? Show your work.
5. Use a Venn Diagram and the given information to determine the number of elements in
the indicated region:
n(U) = 147, n(A) = 48, n(B) = 68, 𝑛(𝐴 ∩ 𝐵) = 19, 𝑛(𝐴 ∩ 𝐶) = 22,
𝑛(𝐴 ∩ 𝐵 ∩ 𝐶) = 10, 𝑛(𝐴′ ∩ 𝐵 ∩ 𝐶 ′ ) = 39, and 𝑛(𝐴′ ∩ 𝐵 ′ ∩ 𝐶 ′ ) = 36.
Find 𝑛(𝐶).
6. You are to create a 5-position password for your e-mail account using the digits 0 to 9
and the letters A though E.
a. How many different codes can be made?
b. If you are only allowed to use a digit or letter once, how many different codes can be
made?
7. Using only the digits 0 and 1, how many different numbers consisting of 8 digits can be
formed?
2
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
8. How many different license plate numbers can be made using 2 letters followed by 4
digits, if
a. Letters and digits may be repeated
b. Letters may be repeated, but digits are not repeated?
c. Neither letters nor digits may be repeated?
9. A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking,
how many ways can you get 4 apples?
10. Three boys and two girls are going to a movie. How many ways can they sit next to each
other under the following conditions:
a. Neither girl sits next to each other
b. The three boys sit next to each other
11. A group of 9 people is going to be formed into committees of 4, 3, and 2 people. How
many committees can be formed if:
a.
a person can serve on any number of committees
b. no person can serve on more than one committee
3
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
12. Draw a tree diagram showing all permutations of size 2 from letters w, x, y, z. How many
permutations are there?
13. Suppose an experiment consists of tossing a coin seven times and recording the
sequences of heads and tails.
a. How many outcomes are possible?
b. How many outcomes have exactly three heads?
c. How many outcomes have at least three heads?
14. Anna goes to a frozen yogurt shop. She can choose from any of the following toppings:
cashews, marshmallow cream, peanut butter chips, blackberries, and brownie bits. How
many different variations of yogurt and toppings can be made, if any combination of
toppings is permitted?
15. In how many ways can a selection of at least one card be made from a hand of five cards?
16. Using the Binomial theorem expand the expression: (𝑥 + 4𝑦)3
4
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
Chapter 6
17. An experiment consists of tossing a coin three times and recording the sequence of heads
and tails.
a. What is the sample space?
b. Determine the event E = “More heads than tails occur.”
c. Determine the event F = “The number of heads equals the number of tails.”
18. Let S = {a, b, c, d, e} be a sample space, E = {a, b, e}, and F = {b, c}.
a. Determine the events 𝐸 ∪ 𝐹 and 𝐸 ′ ∩ 𝐹′.
b. Are 𝐸 ∪ 𝐹 and 𝐸 ′ ∩ 𝐹′ mutually exclusive?
19. What is the probability of getting either a black card or an ace in one draw from an
ordinary deck of 52 cards?
20. The Chicago Black Hawks hockey team has a probability of winning of 0.6 and a
probability of losing of 0.25. What is the probability of a tie?
21. If it has been determined that the probability of an earthquake occurring on a certain day
in a certain area is 0.02, what are the odds of an earthquake occurring?
5
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
22. In a class of twenty students, what is the likelihood that two or more have the same
birthday? (Ignore leap years and assume that each of the 365 days are equally likely).
23. The probability that a person passes organic chemistry the first time he enrolls is 0.8. The
probability that a person passes organic chemistry the second time he enrolls is 0.9.
a. Find the probability that a person fails the first time but passes the second time.
b. Find the probability that a person fails both times.
24. In a small town it is known that 20% of the families have no children, 30% have 1 child,
20% have 2 children, 16% have 3 children, 8% have 4 children, and 6% have 5 children
or more. Find the probability that a family has more than 2 children if it is known that it
has at least 1 child.
25. Seventy percent of the students enrolled in a calculus course had previously taken a
precalculus course. Twenty percent of the students who took a precalculus course earned
an A in the calculus course, whereas 10% of the other students earned an A in the
calculus course.
a. Draw a tree diagram summarizing these data and label it with the appropriate
probabilities.
b. Find the probability that a student selected at random did not earn an A in calculus if
he/she did not take a precalculus course.
6
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
c. Find the probability that a student selected at random did not take a precalculus
course and earned an A in calculus.
d. Find the probability that a student selected at random earned an A in calculus.
e. Find the probability that a student selected at random previously took a precalculus
course, given that he/she earned an A in calculus.
26. Three boxes—I, II, and III—contain three red and two green chips, two red and four
green chips, and four red and five green chips, respectively. A box is selected at random
and a chip is drawn at random from the box.
a. What is the probability that the chip is green?
b. Given the chip is green, what is the probability that it came from box II?
27. A local store orders lightbulb from two suppliers, AAA Electronics and ZZZ Electronics.
The local store purchases 30% of the bulbs from AAA and 70% of the bulbs from ZZZ.
Two percent of the bulbs from AAA are defective while 3% of the bulbs from ZZZ are
defective.
a. Find the probability that a bulb is defective.
b. Find the probability that a randomly selected defective light bulb was purchased from
AAA Electronics.
7
Pace University
Fall 2020
MAT 104 Study Guide
for Exam 2
Mathematics Department
28. A hospital billing department knows that the probability patients 60 or older pay the
balance of their bill after one billing is 80%, while for a person under the age of 60 the
probability is 45%. Seventy percent of the hospital’s patients are 60 or older.
a. What is the probability the balance is paid after one billing?
b. The balance of the bill is not paid after one billing. What is the probability the patient
was 60 years or older?
29. A test for a certain drug produces a false negative 5% of the time and a false positive 8%
of the time. Suppose 12% of the employees at a certain company use the drug.
a. If an employee at the company tests positive, what is the probability that he or she
does not use the drug?
b.
What is the probability that a nondrug user at the company tests positive twice in a
row?
c. What is the probability that a company employee who tests positive twice in a row is
not a drug user?
8