University Problem Set #5 ECE 313
of Illinois Page 1 of 2 Spring 2003
Assigned: Wednesday, February 19
Due: Wednesday, February 26
Reading: Yates and Goodman: Chapters 1, 2 and 3.6
Noncredit Exercises: (Do not turn these in) p. 20: Quiz 1.5; p. 27: Quiz 1.7; p. 79: Quiz 2.9;
pp. 38-39: Problems 1.5.1 – 1.5.6, p. 86: 2.9.1 – 2.9.8
Problems:
1. A long message is divided into L packets of N bits each (including headers, address, CRC
bits, tail, flags, etc. etc. etc.), and transmitted over a channel with bit error probability p. If
the CRC detects that a packet is received in error, the packet transmission is repeated. But,
if a packet has been transmitted a total of five times, and has not been received correctly
even on the fifth try, it is not transmitted again, and is deemed to be lost.
(a) What is the probability that a packet is received correctly?
(b) Let Xi denote the number of times that the i-th packet is transmitted over the channel. What
are the possible values of Xi ? What is the pmf of Xi ? (Be careful about P{Xi = 5}!!).
(c) What is the average number of times that the i-th packet is transmitted? i.e., what is E[Xi]?
(d) What is the probability that all L packets are received successfully
= P{entire message is received successfully}?
2. This problem on conditional probability has three unrelated parts.
(a) If P(A|B) = 0.3, P(Ac|Bc) = 0.4, and P(B) = 0.7, find P(A|Bc), P(A), and P(B|A).
(b) If P(E) = 1/4, P(F|E) =1/2, and P(E|F) = 1/3, find P(F).
(c) If P(G) = P(H) = 2/3, show that P(G|H) ≥ 1/2.
3. Let X denote a negative binomial (or Pascal) random variable with parameters (r, p).
Then, X counts the number of trials required to observe r successes where the probability
of success on any trial is p. Given that X = n, what is the conditional probability that the
i-th trial resulted in a success? To avoid trivialities, assume that n > r and also that n > i.
4. [“I’m leaving on a jet plane, Don’t know when I’ll be back again…”] We return to Problem 6 of
Problem Set #4 where 105 passengers hold reservations for a 100–seat flight from Chicago
to Champaign. Now suppose that 15 passengers are arriving in Chicago on a connecting
flight that is on time with probability 1/3 and late with probability 2/3. If the connecting
flight is on time, all 15 passengers show up for the flight to Champaign (no one stops off
in a bar for a drink!); else they all are not there. The other 90 passengers each decide
independently with probability 0.9, as before, to show up for the flight.
(a) Given that the connecting flight is on time, what is the (conditional) probability that all
passengers who show up get seats? Repeat for the case when the connecting flight is late.
(b) Given that everyone who shows up gets a seat, what is the (conditional) probability that the
connecting flight was on time?
5. An urn contains 6 red balls and 4 green balls. A ball is drawn at random from the urn, and
then another ball is drawn at random without the first one being replaced in the urn.
(a) Let R1 denote the event that the first ball is red and R2 the event that the second ball is red.
What is P(R1)? What is P(R2|R1)? What is P(R2|Rc
1 )?
(b) Use the numbers found in part (a) to compute P(R2). Does the answer surprise you?
(c) Repeat parts (a) and (b) for an experiment in which after the first ball is drawn, it is put
back into the urn along with 3 more balls of the same color before the second ball is drawn.
6. Monty Hall, the host of the TV game show “Let’s Make A Deal™”, shows you three
curtains. One curtain conceals a valuable prize, while the other two conceal junk. All three
