University Problem Set #3 ECE 313
of Illinois Page 1 of 2 Fall 2001
Assigned: Wednesday, September 5
Due: Wednesday, September 12
Reading: Ross, Chapters 2.1–2.5, 4.1, 4.3–4.5, 4 .7
Noncredit Exercises: (Do not turn these in) Chapter 4, Problems: 2, 7, 13, 28, 35, 39, 40-43;
Theoretical Exercises: 11, 13, 15.
Problems:
1.(a) Events A and B are disjoint events defined on a sample space.
What is the probability that at least one of the two events A and B did not occur?
If the answer cannot be determined from the given information, say so explicitly.
(b) Events C and D are events defined on a sample space. If the probability that at least one of
the two events occurred is 0.6, and the probability that at least one of the events did not
occur is 0.8, what is the probability that exactly one of the events C and D did not occur?
If the answer cannot be determined from the given information, say so explicitly.
2. A certain town has three newspapers A, B, and C. The proportions of townspeople that
read these newspapers are as follows:
A: 10%, B: 30%, C: 5%; A and B: 8%, A and C: 2 %, B and C: 4%,
and 1% read all three newspapers.
(a) What percentage of people read only one newspaper?
(b) What percentage of people read at least two newspapers?
(c) If A and C are morning papers and B is an evening paper, what percentage of people read
at least one morning paper as well as the evening paper?
(d) How many people do not read any newspapers?
3. The experiment consists of picking a student from the set of all UIUC students registered
this semester. It is not necessary to assume that all students are equally likely to be picked,
but you may make this assumption if it makes you feel happier and more confident.
(a) Let A and B denote the events that the student picked has had respectively four years of
science (FYS) and calculus in high school. Let P(A) = 0.45 and P(B) = 0.35. If the
probability that the student had neither FYS nor calculus is 0.3, what is the probability that
the student had both FYS and calculus? What is the probability that the student had FYS
but not calculus ?
(b) Let C denote the event that the student is registered in ECE 313, and let A and B be as in
part (a). Suppose that P(A∩B∩C)=0.002. What is the probability that the student picked
is not registered in ECE 313, but did have both FYS and calculus ? If the probability that
the student picked is registered in ECE 313, and has had either FYS or calculus (but not
both) is 0.004, and if students who had neither FYS nor calculus did not register in ECE
313, what is P(C)?
(c) Using the data given in parts (a) and (b), which of the following probabilities can you
compute? It is not necessary to actually compute each probability.
P(A ∪ C), P(A∪B∪C), P(A ∪ B∪ Cc), P(AcBcCc), P(AcBCc), P(ABCc)
4. Let A, B, C denote three events defined on a sample space. Show that
P(A) + P(B) + P(C)
3 ≤ P(A ∪ B ∪ C) ≤ P(A) + P(B) + P(C)
5. Find P(A ∪ (Bc ∪ Cc)c) in each of the following four cases:
(a) A, B, and C are mutually exclusive events and P(A) = 1/3.
(b) P(A) = 2P(BC) = 4P(ABC) = 1/2.
(c) P(A) = 1/2, P(BC) = 1/3, and P(AC) = 0.
(d) P(Ac ∩ (Bc ∪ Cc)) = 0.6.
6. Borrowing an idea from the Big Ten, the NCAA Basketball Tournament has five teams in
its semifinal round (Final Four). Each team plays the other four teams exactly once in the
semifinal round.
