Intro
to Probability and Statistics
Sample
Midterm #3 – Questions Only
Professor Brian Shydlo
Question
1) (12 points) My
friend, Marianne, likes to go to Saratoga and bet on the horses.
She would like
to bet on the trifecta (also known as the triple) in the first race. In the trifecta, you pick, in order, the
first three horses in the race. There
are 8 horses in the first race.
Question
1a) (3 points) In order to be certain of winning, Marianne would like to bet on
all the trifectas. How many different
bets must she place?
Answer: __________________
Question
1b) (3 points) She has $500 dollars.
Each bet is $2 each. Does she
have enough money to bet on all the trifectas?
Answer: __________________
Question
1c) (3 points) To box a trifecta means to pick the first three horses of a
race, without regard to the order in which they place. How many different boxed trifecta bets are
possible for the first race?
Answer: __________________
Question
1d) (3 points) An exacta (also known as perfecta) is when you pick the first
two horses in a race in order. How much
money will Marianne have left, if any, is she bets all of the possible boxed
trifectas AND bets all the possible exactas? Remember that there are 8 horses, each
bet is $2, and Marianne has $500.
Answer: __________________
Question
2) (12 points) My
friend, Jeannie, is an excellent hostess and often invites friends over to her
apartment. When she invites people over,
she always orders Chinese food. She
always orders 8 rolls, 50% spring rolls and 50% Egg Rolls.
Question
2a) (3 points) My friend, Seth, asks me to get him a spring roll from the
kitchen. I really can't tell the
difference, so I just grab a roll at random and bring it back to Seth. What are the odds that Seth gets the roll
that he wants? (Seth can tell the
difference)
Answer: __________________
Question
2b) (3 points) Suppose I want to bring back enough rolls to ensure that Seth
gets the spring roll that he asked for.
(I'll bring the rest back to the kitchen after he selects the spring
roll he wants.) What is the fewest
number of rolls that I need to bring to ensure that Seth gets a spring roll?
Answer: __________________
Question
2c) (3 points) Suppose I can only carry back three rolls from the kitchen. What are the odds that AT LEAST one of the
three is a spring roll?
Answer: __________________
Question
2d) (3 points) Suppose I can only carry back three rolls from the kitchen. What are the odds that EXACTLY one of the
three is a spring roll?
Answer: __________________
Question
3) (23 points) My
friend, Kaplan, is an excellent student.
The odds of him getting an A in any class he takes are 95%. There is a 5% chance of him getting a B. There are no other grades he can get. He is in the MBA program at Stern. There are 20 classes that he'll take to
graduate.
Question
3a) (4 points) What are the odds that Kaplan will graduate with a 4.00 average? Assume that the grade he gets in each class
is INDEPENDENT.
Answer: __________________
Question
3b) (4 points) What is the expected number of As that Kaplan will get?
Answer: __________________
Question 3c)
(4 points) My friend Rich is also very smart and is also in the MBA
program. He has a 90% chance of getting
an A in any given class. What are the
odds that Rich will get a better grade than Kaplan in any single class? Remember, there are only two grades, A and
Not A (otherwise known as getting a B) and everything is INDEPENDENT.
Hint:
Use a Probability Box to get the answer (you don't have to use a box, if you
don't want/need to).
Answer: __________________
Question
3d) (4 points) What are the odds that Rich will graduate with a 4.00
average? Assume that the grade he gets
in each class is INDEPENDENT and that there are 20 classes.
Answer: __________________
Question
3e) (4 points) What are the odds that Rich will graduate with a 4.00 average
AND Kaplan will NOT graduate with a 4.00 Average? Assume independence for everything. Remember they each take 20 courses.
Hint 1:
This can be solved the same way as question 3c.
Hint 2:
I am not asking what are the odds that Rich will graduate with a higher GPA
than Kaplan, which is a different question.
Answer: __________________
Question
3f) (3 points) Assume that at the end of the program Rich has a 4.0 and Kaplan
does not. Should Kaplan be upset? What should Kaplan say if Rich gloats about
his success (not that Rich would)?
Answer:
Question
4) (8 points) You have 4 dice. Each is a normal six-sided die with a 1/6
probability of landing on each of the sides. You roll all 4 die and add up the scores. So the lowest possible score is 4, in which
case you got all 1s and the highest possible score is 24, in which case you
must have rolled all 6s.
Question
4a) (4 points) What is the expected value of the sum of the 4 die?
Answer: __________________
Question
4b) (4 points) What is the Variance of
the sum of the 4 die?
Hint:
The Standard Deviation of a single die is 1.708.
Answer: __________________
Question
5) (21 points)
Assume that you have a portfolio of 7 Bonds in total. You have 3 B rated bonds and 4 D rated
bonds. The following chart will be the
default rate for those bonds. Assume
that all defaults of B relative to D bonds are INDEPENDENT.
B Bonds
|
# Defaults |
Probability |
|
0 |
50% |
|
1 |
35% |
|
2 |
10% |
|
3 |
5% |
D Bonds
|
# Defaults |
Probability |
|
0 |
1% |
|
1 |
5% |
|
2 |
10% |
|
3 |
30% |
|
4 |
54% |
Question
5a) (4 points) What is expected value of the number of defaults for a B bonds?
Answer: __________________
Question
5b) (4 points) What is variance and standard deviation of the number of
defaults for a B bonds?
Answer: __________________
Question
5c) (3 points) What are the odds that you get exactly 0 defaults in your whole
portfolio?
So you get
0 B rated bonds defaulting and 0 D rated bonds defaulting.
Hint: I
am looking for the Intersection
Answer: __________________
Question
5d) (3 points) What are the odds that you get 0 B rated bond defaults or 0 D
rated bond defaults (or 0 total defaults).
Hint: I
am looking for the Union
Answer: __________________
Question
5e) (4 points) What are the odds that you get exactly 1 default in your whole
portfolio?
So you get
1 B rated bond defaulting and 0 D rated bonds defaulting OR
So you get
0 B rated bond defaulting and 1 D rated bonds defaulting.
Hint:
Create a probability box
Answer: __________________
Question
5f) (3 points) Assume that you have a portfolio of 7 Bonds in total. You have 3 B rated bonds and 4 D rated
bonds. (same as before so far) Now assume that the bond defaults are
binomially distributed.
pB-Bond
= the probability that a B rated bond defaults = 20%
pD-Bond
= the probability that a D rated bond defaults = 90%
Fill in the
chart using the binomial distribution odds:
B Bonds
|
# Defaults |
Probability |
|
0 |
|
|
1 |
|
|
2 |
|
|
3 |
|
D Bonds
|
# Defaults |
Probability |
|
0 |
|
|
1 |
|
|
2 |
|
|
3 |
|
|
4 |
|
Question
6) (24 points) My
friend Tony has a width of 15 inches.
(Don't confuse width with waist size, which would be diameter x pi = (15
inches * 3.14) if Tony's waist was a perfect circle.)
A person's
width is normally distributed with a mean of 12 inches and a standard deviation
of 1.5 inches.
The seats
at a Broadway show are exactly 14 inches wide.
Anyone who comes to a show who is greater than 14 inches wide will be
uncomfortable. Anyone who comes to the
show who is exactly 14 inches or less will be comfortable.
Question
6a) (3 points) Suppose Tony goes to the theater above. What are the odds that he will be
uncomfortable?
Answer: __________________
Question
6b) (6 points) A random person comes into the theater. This person's width follows the normal
distribution as described above. What
are the odds that this person will be uncomfortable?
Answer: __________________
Question
6c) (6 points) A person is ultra-comfortable if they have 3 or more inches of
extra room in their chair. Assume that a
random person comes into the theater.
This person's width follows the normal distribution as described above
and the chairs are 14-inches wide. What
are the odds that this person will be ultra-comfortable?
Answer: __________________
Question
6d) (3 points) Assume that a particular theater on Broadway has 10 rows of
seats, each of the 10 rows is 70 feet across.
Each seat is exactly 14 inches wide and there are no gaps or aisles
between the seats. The theater fills up
every night and they charge $50 per seat.
How much money do they make each night?
Answer: __________________
Question
6e) (3 points) Assume that the owner of the theater can at zero cost switch to
seats that are 15 inches wide. How much
money will they lose relative to the 14 inch wide seats? Assume that they fill up the theater with
either size seat.
Answer: __________________
Question
6f) (3 points) More seats mean more money when you assume that the theater is
always full. That assumption will not
hold if the seats get too small. People
will not be comfortable and not go to your theater.
What sort
of analysis would you do to determine the optimal size of seats in your theater
assuming you want to maximize profits?
What do you tell Tony if he complains that the seats are uncomfortable?
Answer: