University Problem Set #2 ECE 313
of Illinois Page 1 of 3 Summer 2003
Assigned: Thursday, June 19
Due: Thursday, June 26
Reading: Ross Chapters 2.1-2.5, 4.1-4.6
Noncredit Exercises: (Do not turn these in) Ross Chapter 1: Problems 1-5,7,9; Theoretical Exercises 4,8,13;
Self-Test Problems 1-15. Chapter 2: Problems 3,4,9,10,11-14; Theoretical Exercises 10,20; Self-Test Problems 1-
17.
1. Ross, Problem 6, page 59 (6th edition),or page 62 (5th edition), or page 55 (4th edition).
2. Consider an experiment with a finite sample space containing n equally likely outcomes.
Thus, there are 2n different events defined on this sample space.
(a) Show that 2n-1 events are comprised of an odd number of outcomes while 2n–1 events are
comprised of an even number of outcomes. (Zero is an even number)
(b) Find the “average probability” of an event by adding up the probabilities of all 2n events
and dividing the resulting sum by 2n.
(c) How many of the 2n different events have probability equal to the average probability
that you found in part (b)?
(d) It was noted in class that when a trial of the experiment is performed, exactly 2n–1 events
occur while the other 2n–1 events do not occur. What is the average probability of the 2n–1
events that do occur on a given trial of the experiment?
If you cannot solve this problem for general n, solve it (for 50% credit) for the case
n = 3 and Ω = {a,b,c}. If you choose to solve only this special case, then, in part (d),
assume that outcome b occurs.
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. An experiment consists of observing the contents of an eight-bit shift register. Assume
that all 28 = 256 bytes are equally likely to be the contents of the shift register.
(a) Let A denote the event that the least significant bit in the shift register is a 1. What is
P(A)?
