Economic Statistics: Problem Set 1
Vladyslav Nora
Fall 2024
1
Combinatorics
1. Problems after chapter 1 in BH book: 1 - 7, 10, 24, 28, 30, 39, 42, 47.
2. Attached is the list of names of students in our class. Note that names repeat because some
people have the same name.
(a) What is the probability that a name of a randomly selected student starts with an A?
(b) What is the probability that a name of a randomly selected student starts with an A or B?
(c) What is the probability that a name of a randomly selected student starts with an A and
does not end with an L?
(d) Consider randomly selecting a student. What is the probability that there is no other
student with the same name?
(e) Randomly select two students. What is the probability that they share the same name?
(f) Randomly select three students. What is the probability that they don’t share the same
name?
(g) Now lets try to validate your answers using a simulation. You will have to select 2 students
randomly 100 or more times. Then compute a proportion of times when the students share
the same name. We expect this number to be “close” to your theoretical answer. To complete
this assignment you can use simply use any spreadsheet and a random number generator
in excel or online. When selecting two students, you must make sure that each pair can be
selected with equal chance. You can also use R or any other software. Describe the method
you have used and the results.
(h) What do you think is the probability that by the end of a semester two top students in
this class will have the same name? Carefully explain the assumptions you make while
answering this question.
3. McDonald’s in Mega Silk Way ran a lottery. Each customer received a scratch ticket that had
11 covered spots, containing 3 burger, 3 Cola, 3 fries, and 2 McDonald’s symbols. Below is an
example of a ticket with all the spots scratched off.
1
(a) How many ways are there to scratch off a single spot such that there is a Cola symbol
behind it?
(b) How many ways are there to scratch off two spots such that there is a Cola symbol behind
each scratched spot?
(c) How many ways are there to scratch off three spots such that there is a Cola symbol behind
each scratched spot?
(d) How many ways are there to scratch off two spots such that there is no Cola behind each
scratched spot?
(e) How many ways are there to scratch off three spots such that there is no Cola and no Fires
behind each scratched spot?
(f) You randomly choose a single spot to scratch off. What is the probability that there is a
Cola symbol behind it?
(g) You randomly choose three spots to scratch off. What is the probability that there is a Cola
symbol behind each of them?
(h) You randomly choose three spots to scratch off. Assume that you win, if the three symbols
behind the scratched off spots coincide. What is the probability to win? You can leave your
answer as a fraction of binomial coefficients.
(i) You randomly choose three spots to scratch off. Now assume that McDonald’s symbol can
be used to substitute any other symbol (like a joker in poker). What is the probability to
win? You can leave your answer as a fraction of binomial coefficients.
4. Consider the figure below.
2
(a) How many rectangles are there in the figure (a rectangle can consist of several smaller
rectangles)?
(b) Generalize you result for any m × n initial grid (for example the figure above contains 6x4
grid).
(c) How about the figure below? Find a systematic way to count for such an incomplete
rectangle (which misses k × l rectangle in the upper right corner, k < m and l < n).
2
Conditional probability
1. Problems after chapter 2 in BH book (2nd edition): 3, 6, 8, 13, 16, 20, 33.
2. A coin is tossed three times. What is the probability that exactly two heads occur, given that
(a) the first outcome was a head?
(b) the first outcome was a tail?
(c) the first two outcomes were heads?
(d) the first two outcomes were tails?
(e) the first outcome was a head and the third outcome was a head?
3. Maja and Oleg play a game. Initially they start with some quantity of soldiers each. Each round
they have a battle where a probability to win a battle for a player with k soldiers against an
k
opponent with l soldiers is k+l
. If a player wins a battle, then he or she takes one soldier from the
opponent. A player who rans out of soldiers looses (and the other one wins). Find a probability
that Maja wins the game when she initially has 2 soldiers and Oleg has only 1.
4. A prisoner is given a chance for parole. He’s blindfolded and told to choose one of two bags;
once he does, he is to reach in and pull out a marble. Each bag has 25 red and 25 blue marbles,
and the marbles all feel the same. If he pulls out a red marble he is set free; if it’s a black, his
parole is denied.
(a) What is his chance of winning parole?
(b) Now the prisoner is free to distribute the marbles among the two bags however he wishes, so
long as all the marbles are distributed. After he distributes the marbles, he is blindfolded,
and chooses a bag at random and then a marble. Find the highest probability of being set
free the prisoner can achieve.
3
Probability distributions
1. Problems after chapter 3 in BH book (1st edition): 1, 2, 4, 5, 15, 17, 20, 38, 41.
2. Suppose that a random variable X has the discrete uniform distribution on the integers 10, . . .
, 20. Find the probability that X is even.
3. Suppose that two balanced dice are rolled, and let X denote the absolute value of the difference
between the two numbers that appear. Determine and sketch the p.m.f. and c.d.f. of X.
4. In order to find out how many fish there are in a lake, 1000 fish are caught, marked and put back
into the lake. For what number of fish in the lake will the probability be greatest of encountering
10 marked fish among 150 fish caught a second time
3