Probability Theory

1. There are two events A and B, both with nonzero probabilities. If the occurrence of
B makes occurrence of A more likely, is it ALWAYS true that the occurrence of A also
makes occurrence of B more likely? (hint: "B makes occurrence of A more likely" can be
represented as "P(A│B) > P(A)")

2. A parking lot consists of a single row containing n parking spaces (n ≥ 2). Mary arrives
when all spaces are free. Tom is the next person to arrive. Each person makes an equally
likely choice among all available spaces at the time of arrival.

(a) Describe the sample space.
(b) Let A be the event that the parking spaces selected by Mary and Tom are at most
2 spaces apart (that is, at most 1 empty space between them). What is P (A)?

3. A large ISP ran an experiment over 1 year to determine the predominance of spam email
on their servers. It turns out that of all the mails they have received, 20% are junk mails.
Assume that for each email you receive, there is also a .20 probability of it being spam.
Furthermore, assume that the probability of one email being spam is independent of the
probability of any previous email being spam. If you have received 10 mails during a
given day, what is the probability that you have received:

(a) At least one junk mail?
(b) Exact one junk mail?
(c) At least two junk mails?

4. Three manufacturers (Acme,Binford and Craftsmen) make the same product with the
following rates:

_ Acme produces 40% of the total output with a non-defective ratio of 0.85
_ Binford produces 35% of the total output with a non-defective ratio of 0.90;
_ Craftsmen produces 25% of the total output with a non-defective ratio of 0.95.

A customer receives a defective product. What is the probability that it came from
Binford? What assumptions did you make to answer this question?

5. Assume Internet traffic consists of 50% TCP packets, 20% UDP packets, and 30% ICMP
packets. What is the probability that when we randomly pick 5 packets from the traffic,
two are UDP packets, one is TCP packet, and two are ICMP packets? What assumptions
did you make to answer this question?