1. Compute the mean, median, Q1 , Q3 and the standard deviation for the following
observations:4, 5 ,7 , 12 ,5, 8.
2. If the mean of 20 observations is 32, then what is the sum of all of the observations?
3. A student received scores of 92, 83, and 71 on three quizzes. What is the lowest
score that the student can get on the next test to achieve a mean of at least 80?
4. If your scores of the first four exams are 98, 100, 90 and 97, what do you need
to make on the next exam for your overall mean to be at least 90?
5. A student received the following grades on quizzes in a history course: 80, 88,
75, 93, 79, 95, 75, and 96.
Find the mean, and median, and the first and third quartiles of this data. Also
compute the standard deviation.
6. Suppose a clown flips a coin until it lands heads up, and then records how many
flips it takes. Write down the sample space for this experiment.
7. Suppose you are watching an intersection at which three cars enter and then
turn either left or right, or go through. You record the sequence of directions
that each of the three cars take (denote a left turn with L, a right turn with R,
and going through with T).
(a) Write down the sample space for this experiment. How many elements
does it have?
(b) Write down the event A that at least one of the cars goes through the
intersection in terms of the outcomes in the sample space.
(c) Write down the event B that that exactly one car turns right in terms of
the outcomes in the sample space.
(d) Write down the following in terms of the outcomes in the sample space.
i. Ac
ii. A ∩ B
iii. B c ∩ (A ∪ B)
(e) Suppose that when a car enters the intersection, it is equally likely to go
in any one of the three directions. Compute P (A) and P (B). You may
assume that the cars turn independently of one another.
(f) Compute P (A|B) and P (B|A)
8. Draw a Venn Diagram which represents the following set:
((A ∩ D) ∪ (B ∩ D)) ∩ (A ∩ B ∩ D)c
9. You have 20 distinct books, 7 of which written by Mark Twain, and another is
Great Expectations by Dickens. You also have a bookshelf which has space for
12 books.
(a) How many ways can you arrange 12 books on the shelf (order matters)?
(b) How many ways can you arrange 12 books on the shelf if exactly 3 of them
must be Mark Twain books (HARD, i.e. Extra Credit Type Problem)
(c) A monkey picks books at random from the 20 books and puts them on the
shelf until it contains 12 books. What is the probability that at least 3 of
the books on the shelf are written by Mark Twain?
(d) What is the probability that the book in the first spot on the shelf (from
left to right) is Great Expectations?
10. Kim coaches her son Mark’s little league team, which has 13 players.
(a) How many ways can Kim make a roster for the team, given that there are
9 positions to fill?
(b) Mark is the teams best player, and therefore should play catcher. How
many ways can Kim make the roster so that Mark is catcher?
(c) Kim has a terrible migraine headache before the game, so she just writes
down players at random to fill the roster. If all possible rosters are equally
likely, what is the probability that Mark plays catcher, and Felix, the teams
best pitcher, pitches?
11. Jack is a pole vaulter, which can be a fairly dangerous sport. On each vault,
his probability of success is 70%. If he succeeds, the chance that he has a
catastrophic injury is only 20%. But if he fails, the probability that he has a
catastrophic injury is 50%.
(a) Compute the probability that Jack has a catastrophic injury.
(b) Compute the probability that Jack failed given that he had a catastrophic
injury.
(c) Compute the probability that Jack succeeded, given that he was ok after
the vault.
12. A coin is flipped 3 times independently. The probability of getting a heads is
.34.
(a) Compute the probability of getting exactly 2 tails in this experiment.
(b) Suppose X = Number of heads. Compute the pmf and the cdf of X.
(c) Compute the expected value and variance of X.
13. Suppose a RV X has pmf p(x) =
c
x2
for x = 1, 2, 3, 4.
(a) Compute the value of c so that p(x) is a bonafide pmf.
(b) Compute the expected value and the variance of X.
(c) Compute the expected value and variance of Y = 4X 2 + 2
14. Additional Book Questions: 2.1; 4 - 2.2; 16, 17 - 2.3; 29, 31 -2.4;61,62 -3.2;
17,20 -3.3;36, 38