Homework 2
ECON 451 Intermediate Introduction to Stats. and Econometrics I
Note: You need to show your work and sufficient effort to receive full completion points. Most questions
are taken from textbook: Larsen and Marx (2018)
1. Prove that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)) (the other distributive property not proved in class).
2. Prove one of DeMorgan’s Law: (A ∪ B)c = Ac ∩ B c
3. Express the following probabilities in terms of P (A), P (B), and P (A ∩ B) only.
• P (Ac ∪ B c )
• P (Ac ∩ (A ∪ B))
4. Events A1 and A2 are such that A1 ∪ A2 = S and A1 ∩ A2 = ∅. Find p2 if P (A1 ) = p1 , P (A2 ) = p2 ,
and 2p1 − p2 = 12
5. If P (A) = a and P (B) = b, prove that:
P (A|B) ≥
a+b−1
b
6. Nine students, five men, and four women interview for four summer internships sponsored by the city
newspaper.
• In how many ways can the newspaper choose a set of four interns?
• In how many ways can the newspaper choose a set of four interns if it must include two men and
two women in each set?
• How many sets of four can be picked such that not everyone in a set is of the same sex?
7. (Bayes Theorem Question) Suppose a cancer diagnostic test is 95% accurate. Assume that 0.005
of the population actually has cancer. What is the probability that the individual has cancer, given
the test says he has cancer?
8. Suppose that P (A) =
1
4
and P (B) = 81 .
(a) What does P (A ∪ B) equal to if i ) they are mutually exclusive, ii ) they are independent
(b) What does P (A|B) equal to if i ) they are mutually exclusive, ii ) they are independent
9. Andy, Bob, and Charley have all been serving time for grand theft auto. According to the prison
warden, two will be released next week. All three have a similar record, thus two will be chosen
randomly. Each person has a 2/3 probability of being included in the two to be set free. Andy is
friends with the prison guard. The guard offers to tell Andy ahead of time one person, other than
himself, who will be released next week. Andy refuses this offer, believing, if he knows, then his
probability of being the other person released drops to 1/2 (since two person is left for 1 spot).
• Is Andy’s concern justified?
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