Intro
to Probability and Statistics
Sample
Midterm #2 – Questions Only
Professor Brian Shydlo
Your Name:
____________________________________________________
Question
1) (18 points in total) Assume
there are 130 MBA students in a given program.
78 are taking an elective on Venture Capital and 40 are taking an
elective on Corporate Governance.
Question
1a) (3 points)
Assuming Statistical Independence between taking Venture Capital and Corporate
Governance, how many students are taking both classes?
Hint:
Intersection
Answer: __________________
Question
1b) (3 points)
Again, assuming Statistical Independence between taking Venture Capital and
Corporate Governance, how many students are taking at least one of the two
courses (either one class or the other or taking both)?
Hint:
Union
Answer: __________________
Question
1c) (3 points) Now
assume that these classes are not Statistically Independent. It turns out, if a student is taking Venture
Capital, then it is less likely that they are taking Corporate Governance.
130
MBA students in a given program.
(Same as before.)
78 are
taking an elective on Venture Capital. (Same as before.)
40 are
taking an elective on Corporate Governance.
(Same as before.)
Assume 10%
are taking both. (new)
How many
students are taking exactly one of the two courses?
Answer: __________________
Question
1d) (3 points) How many students are not taking
either class? Please use the same assumptions and numbers as part c.
Answer: __________________
Question
1e) (3 points) What is the probability that a student is
taking Venture Capital given that the student is taking Corporate
Governance?
Hint: What
is P(A | B)?
Please use
the same assumptions and numbers as part c.
Answer: __________________
Question
1f) (3 points) Now assume you have 130 students in an MBA
program and 78 are taking Venture Capital and 68 are taking Credit Risk
Management. Are these classes Mutually
Exclusive? Why or why not?
Answer:
________________________________________________________
Question
2) (15 points in total)
You own 1 stock with the following distribution of returns:
|
Kind of Year |
Probability Of This Kind Of Year
Occurring |
Stock Return |
|
Great |
10% |
35% |
|
Good |
25% |
20% |
|
Fair |
45% |
15% |
|
Poor |
15% |
5% |
|
Worst |
5% |
-30% |
Question
2a) (6 points) What
is the Expected Value of your stock return?
Answer: __________________
Question
2b) (6 points) What
is the Variance of your stock return?
Answer: __________________
Question
2c) (3 points) What
is the Standard Deviation of your stock return?
Answer: __________________
Question
3) (31 points in total)
You manage Za House, a trendy new pizza place (Za as in pizZa). Your restaurant delivers and it has a
guarantee regarding how long it takes to deliver a pizza. Your policy is that you will deliver a pizza
in 30 minutes or less or it is free.
If you
deliver the Pizza in 30 minutes or less then you make 2 dollars
If it takes
more than 30 minutes you lose 10 dollars
You deliver
exactly 10 pizzas every day. These are
all to the same place, to the MBA students at NYU, who like their pizza. Each pizza is delivered as a separate trip,
so you make exactly 10 trips per day.
Assume each pizza delivery is independent of the other ones (which
normally would not be the case, unless you had 10 pizza delivery people, but we
have simplified things for this question).
You have
carefully mapped out the shorted (in distance) route, but you are concerned
because you are still giving out free pizzas from time to time. You are considering either stopping the free
deliveries or switching to a different route.
You are not sure what to do, as any other route you take would be longer
in distance. You carefully examine two
possible routes. Route A is the one you
are taking now. Route B is another way that is longer in distance.
Please
assume for this problem that Route A is normally distributed with a mean of 26
minutes and a standard deviation of 3 minutes.
(Ignore the fact that delivery times can't be negative, so the Normal
Distribution is not a perfect fit.
Please assume it is Normally Distributed for the purposes of this
problem.)
Please
assume for this problem that Route B is normally distributed with a mean of 27
minutes and a standard deviation of 2 minutes.
(Ignore the fact that delivery times can't be negative, so the Normal
Distribution is not a perfect fit.
Please assume it is Normally Distributed for the purposes of this
problem.)
|
Route |
μ (the mean) |
σ
(the Standard Deviation) |
|
A |
26
minutes |
3 minutes |
|
B |
27
minutes |
2 minutes |
Question
3a) (4 points) Why
might the Standard Deviation of Route A be higher than the Standard Deviation
of Route B?
Answer:
__________________________________________________________
Question
3b) (4 points) What
is the probability that a single delivery of pizza using Route A will
result in a free delivery? (Or what is
the probability that the time will take more than 30 minutes)?
Answer: __________________
Question
3c) (4 points) What
is the probability that a single delivery of pizza using Route B will
result in a free delivery? (Or what is
the probability that the time will take more than 30 minutes)?
Answer: __________________
Question
3d) (3 points) What
is the expected value of the profit of Route A for a given day (remember there
are 10 deliveries in a day)?
Answer: __________________
Question
3e) (3 points) What
is the expected value of the profit of Route B for a given day (remember there
are 10 deliveries in a day)?
Answer: __________________
Question
3f) (3 points) Which route would you choose to use and why? Would you choose Route A or
Route B?
Answer:
_________________________________________________________
Question
3g) (4 points)
Looking back at Route A, please write down the distribution of possible
outcomes and total payoffs for the day (without regard to probabilities).
Hint:
There are 11 possible outcomes for the day.
With 0 trials over 30 minutes you make 10 * 2 = $20.
Answer:
_________________________________________________________
Question
3h) (6 points) What
are the odds of having a profitable day using Route A? (Or what are the odds of having a day with a
profit of more than 0 using Route A?)
Hint:
Each trial is assumed to have a normal distribution as given in the intro to
this problem. Each trial is independent
from every other trial and there are only two possible outcomes for each trial,
either you are over or under 30 minutes.
If you didn't figure out the probability of the time being more than 30
minutes for Route A, then just use any number for the probability between 5%
and 25% for the odds of going over 30 minutes on one trial and if you get a
logically consistent answer you'll get full credit.
Answer: __________________
Question
4) (6 points) A
random variable X is Normally distributed with a mean
of 10. The probability that x>12 is
15.87%. What is the odds that x is
between 9 and 11?
X is n(10,
???): X is normally distributed with a mean
of 10 and a standard deviation you need
to figure out.
p( 9 < X
<= 11)?
Answer: __________________
Question
5) (10 points in total) The average American adult man has a
height that is normally distributed with a mean of 67 inches and a Standard
Deviation of 6 inches.
The average
American adult woman has a height that is normally distributed with a mean of
63 inches and a Standard Deviation of 4 inches.
Question
5a) (5 points) How
big does a bed maker need to make a bed (in inches) to accommodate 90% of all
Adult Male Americans?
Answer: __________________
Question
5b) (5 points) What percentage of American Females fits on a bed the size
of the answer to part "a"?
Answer: __________________
Question
6) (20 points in total) A particular car has 6 sub systems. Each
system must be working for the car to run (for the car to be operable). The probability of a system failing is
10%. The probability of a system working
(not failing) is 90%. Each system is completely independent of the other
systems.
Question
6a) (4 points) What
is the expected number of failures in the car?
Meaning for all 6 systems, how many are expected to fail on average?
Answer: __________________
Question
6b) (4 points) What
are the odds that the car will run? (Or
what are the odds of zero subsystem failures of the 6 subsystems)?
Answer: __________________
Question
6c) (4 points) What would the per sub-system working
rate need to be in order to have the total success rate for the car to be
operational be 99%? Said a different
way, what would the odds of working need to be for each sub-system to expect
that the odds for all 6 system working (no failures of any subsystems) at the
same time to be 99%?
Hint:
This relates to question 6b. The answer
involves doing a n-square-root. For those without a calculator, I have
included the table below. You would read
it by taking the first column to the power of the top row and the intersecting
cell is the answer. For example, the cube root of .01 is 0.215443 (or .011/3
= .215443).
|
|
1/2 |
1/3 |
1/4 |
1/5 |
1/6 |
|
0.01 |
0.1 |
0.215443 |
0.316228 |
0.398107 |
0.464159 |
|
0.05 |
0.223607 |
0.368403 |
0.472871 |
0.54928 |
0.606962 |
|
0.1 |
0.316228 |
0.464159 |
0.562341 |
0.630957 |
0.681292 |
|
0.9 |
0.948683 |
0.965489 |
0.974004 |
0.979148 |
0.982593 |
|
0.95 |
0.974679 |
0.983048 |
0.987259 |
0.989794 |
0.991488 |
|
0.99 |
0.994987 |
0.996655 |
0.997491 |
0.997992 |
0.998326 |
Answer: __________________
Question
6d) (3 points) Now assume that failures are not
Binomially Distributed. Assume the following distributions for numbers of
defect in total.
|
# defects |
Prob of this many
defects |
|
0 |
40% |
|
1 |
40% |
|
2 |
10% |
|
3 |
10% |
|
4 |
0% |
|
5 |
0% |
|
6 |
0% |
What is the
expected value of the number of failures?
Answer: __________________
Question
6e) (3 points) Assume
the same distribution as part d. What is
the Standard Deviation of the distribution?
Answer: __________________
Question
6f) (2 points) Is
the odds of 2 failures (as per the chart of probabilities in section d)
Statistically Independent with the odds of 3 failures?
e.g. let A
= 2 failures
Let B = 3
failures
Are A and B
Statistically Independent?
Answer: __________________