Assignment 1:
Probability and Expected Value
Due in class of week 3.
1. Answer the following statements TRUE or FALSE, providing a succinct explanation of your
reasoning.
(a) If the odds in favor of A are 3:5 then P(A) = 0.4.
False; probability of A is 3/(3+5) = 3/8.
(b) You roll two fair three-sided dice. The probability the two dice show the same number
is 1/3.
True; there are three possibilities, (1,1), (2,2) and (3,3), each with (1/3)(1/3) = 1/9 probability, so the
probability of a match is 1/9+ 1/9 + 1/9 = 1/3.
(c) If events A and B are independent and P(A) > 0 and P(B) > 0, then P(A and B) > 0.
True; P(A and B) = P(A)P(B) if A and B are independent, and the product of two positive numbers is positive.
(d) If two events A and B are independent and P(A) > 0 and P(B) > 0, then A and B
cannot be mutually exclusive.
True; P(A and B) = P(A)P(B) if A and B are independent, and the product of two positive numbers is positive.
(e) If P(A and B)
0.4 then P (A)  0.40.
False; P(A) is greater than or equal to P(A and B) — a more specific event is always less likely than a
more general event of which it is a subpart.
(f) If two random variables have non-zero correlation, then they must be dependent.
True; a non-zero correlation measures the linear dependence between two random variables.
(g) If two random variables have zero correlation, then they must be independent.
False; having no linear dependence does not rule out having other (nonlinear) types of dependence.
(h) If two random variables are independent, then the correlation between them must be
zero.
True; two variables being independent means that they have no form of dependence, including linear dependence,
as measured by correlation.
2. An oil company wants to drill in a new location. A preliminary geological study suggests that
there is a 20% chance of finding a small amount of oil, a 50% chance of a moderate amount
and a 30% chance of a large amount of oil. The company has a choice of either a standard
drill that simply burrows deep into the earth or a more sophisticated drill that is capable of
horizontal drilling and can therefore extract more but is far more expensive. The following
table provides the payoff table in millions of dollars under different states of the world and
drilling conditions
small medium large
standard $20M
$30M $50M
horizontal -$20M $40M $90M
(a) Find the mean payoffs of the two different drilling strategies.
E(standard) = 20(0.20) + 30(0.5) + 50(0.30) = $34M
E(horizontal) = -20(0.20) + 40(0.5) + 90(0.30) = $43M
(b) Find the variance in payoffs of each strategy.
V(standard) = E(X^2) - E(X)^2 = 20^2(0.20) + (30^2)(0.50) + 50^2(0.30) - 34^2 = 124
E(horizontal) = (-20)^2(0.20) + (40^2)(0.5) + (90^2)(0.30) - 43^2 = 1461
(c) Which strategy would you advocate for and why?
(d) How much are you willing to pay for a geological evaluation that would determine with
certainty the quantity of oil at the site prior to drilling?
Hint: construct a third option whose payoff is max(horizontal, standard). Compute its expected value.
3. Cooper Realty is a small real estate company located in Albany, New York, specializing
primarily in residential listings. They have recently become interested in determining the
likelihood of one of their listings being sold within a certain number of days. Based on
historical data, they produced the following figures based on the past 800 homes sold.
Days Listed until Sold
Under $50K
$50-$100K
$100 - $150K
Over $150K
Under 20
50
20
20
10
31-90
40
150
280
30
Over 90
10
80
100
10
Total
100
250
400
50
(a) What is the probability that a randomly selected home is listed over 90 days before
being sold?
(10 + 80 + 100 + 10)/(100+250+400+50) = 200/800 = 1/4.
(b) What is the probability that a randomly selected initial asking price is under $50K?
100/800 = 1/8.
(c) What is the probability of both of the previous two events happening? Are these two
events independent?
If they were independent we would have P(A and B) = (1/8)(1/4) = 1/32;
instead we have P(A and B) = 10/800 = 1/80, so they are not independent.
(d) Assuming that a contract has just been signed to list a home that has an initial asking
price less than $100K, what is the probability that the home will take Cooper Realty
more than 90 days to sell?
(10 + 80)/(100 + 250) = 90/350 = 9/35.
4. One hundred people were asked “Do you smoke?” Their responses are summarized in the
following table.
Yes No
Male 19 41
Female 12 28
Total 31 69
Total
60
40
100
(a) To interpret these numbers as probabilities, what must we do so that they normalize
correctly?
Divide by 100 so they sum to one.
(b) What is the probability that a randomly selected person from this group is a smoker?
31/100 = 0.31
(c) What is the probability that a randomly selected smoker from this group is a female?
12/31
(d) Write down the conditional distribution (in the form of a table) of the variable X which
denotes (smoker) or (non-smoker), given Y , which denotes (male) or (female).
We have P(smoker | Y = male) = 19/60 and P(non-smoker | male) = 41/60.
And, we have P(smoker | Y = female) = 12/40 and P(non-smoker | female) = 28/40.
5. At a certain private college students concentrate in one of three areas: pre-med, pre-law, or
theater-arts. Only 15% of the students are theater-arts majors, with the remaining population divided equally among pre-med and pre-law. The most popular intramural sport at this
college is ultimate frisbee, but participation differs between concentrations. Only 10% of
pre-med students play on a frisbee squad, while 20% of pre-law students do, and 80% of
theater-arts students do.
(a) Suppose that your new summer intern shows up wearing an ultimate frisbee t-shirt with
the school logo on it (which you take as certain evidence that she played ultimate frisbee
there in college). What is the probability that this student was a theater-arts major?
See following page.
(b) Suppose we know that pre-med students go on to have a loan-adjusted average postgraduation salary of $60K, pre-law students go on to have a loan-adjusted average postgraduation salary of $80K and theater-arts students have a loan-adjusted average postgraduate salary of -$5K. Given that we think the student did play ultimate frisbee in
college, what is the student’s expected salary?
See following page.