Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
For a more conceptual review, I recommend starting with the concept check questions at the beginning of the review exercises for each chapter. These questions are included in the Chapter
Summaries document on mycourses.
1. Consider the following sentence: “This semester I am going to learn probability.” Suppose
you choose a word randomly from that sentence.
(i) What is the sample space? How many outcomes are there?
(ii) What is the probability that the word does not contain an “r”?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
2. Suppose that in our class there are 4 students of age 18, 32 students of age 19, and 2 students
of age 21. A student is selected randomly to write their age on the blackboard. I ask the
class: what is the probability that they write “21”? One eager student replies, “The set of
all outcomes is the set {18, 19, 21}. Since there are three possible outcomes, the probability
is 1/3.” Is this correct? Explain why or why not.
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
3. Consider the following experiment: Fred rolls a die, Jane tosses a coin, and Pearl draws a
card randomly from a standard deck of cards, all at the same time.
(i) Describe the sample space. How many outcomes are there?
(ii) What is the probability of the event A = “Fred rolls an odd number and Pearl draws a
diamond”?
(iii) What is the probability of the event B = “either Fred rolls an odd number or Pearl
draws a diamond”?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
4. Suppose we roll a red and a green die. Let A =“the red die shows a 2 or a 5” and B = “the
sum of the two dice is at least 7”. Are A and B independent? Show your answer in two
different ways: once using the definition, and once using conditional probability.
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
5. How many subsets of size four of the set S = {1, 2, 3, . . . , 20} contain at least one of the
elements in {1, 2, 3, 4, 5}?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
6. How many five-digit numbers can be formed from the integers 1, 2, . . . , 9 if no digit can appear
more than twice? (For instance, 41434 is not allowed but 41433 is allowed.)
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
7. Use a Venn diagram to show which if the following is true. Suppose that E ⊂ F . Then
(a) E C ⊂ F C
(b) E C = F C
(c) F C ⊂ E C
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
8. If P(E) = 0.9 and P(F ) = 0.8, show that P(E ∩ F ) ≥ 0.7
Fall 2020
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
9. Suppose A ⊂ B and P(A) = 0.5.
(a) Circle the true statement.
i. P(B) ≥ 0.5
ii. P(B) ≤ 0.5
(b) Circle the true statement.
i. P(B|A) = 1
ii. P(B|A) = 0.5
iii. P(B|A) = 0
Fall 2020
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
10. Let X be a random variable and suppose that P(X = 0) = 1 − P(X = 1). If E[X] = 3V ar(X),
find P(X = 0).
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
11. If X is a Binomial random variable with expected value 6 and variance 2.4, find P (X = 5).
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
12. It is impossible to have a random variable X with E[X] = 3 and E[X 2 ] = 8. Use the formula
for V ar(X) to show that this cannot happen.
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
13. Santa Claus has 45 drums, 50 cars, and 55 baseball bats in his sled. 15 kids will get a drum
and a car, 20 a drum and a bat, 25 a bat and a car, and 5 will get all 3 presents.
(a) How many kids will get presents?
(b) How many kids will get just a drum?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
14. You decide to go to the pond to try to capture a frog to give to each of your 2 siblings for
Christmas. You have a 12 probability of catching at least 1 frog, and you are three times more
likely to catch only 1 frog than you are to catch 2 frogs. Being a friend to all creatures, you
decide to never catch more than the 2 frogs you need. Let F be the number of frogs you catch.
(a) What is the probability mass function of F ?
(b) Compute V ar(F ), the variance of F .
(c) Assuming that each of the frogs near the pond have all 4 legs intact, what is the expected
number of frog legs you take home with you?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
15. Suppose X is a normal random variable with mean 10 and standard deviation 4 and Y is a
random variable with mean −2 and standard deviation 20.
(a) Find P(X ≥ 10).
(b) Use a z-table to find P(−3 ≤ Y ≤ 14)
(c) Sketch the probability density functions of X and Y . Where is the peak (highest point)
of each function located? Which is higher: the peak of the probability density function
of X or the peak of the probability density function of Y ?
Math 123 - Beckman
Final Exam Review, Chapters 5, 7-9
Fall 2020
16. Suppose that at the end of every year for 10 years, you put $ 100 into an account with 4.8%
interest, compounded monthly.
(a) What is the future value of the account after 10 years?
(b) What is the present value of the account?
(c) How much interest in earned over the course of the 10 years?