University Problem Set #2 ECE 313
of Illinois Page 1 of 2 Fall 1997
Assigned: Wednesday, September 10, 1997
Due: Wednesday, September 17, 1997
Reading: Ross, Chapter 2.1-2.5 and 2.7, Chapter 3
Noncredit Exercises: (Do not turn these in) Ross pp. 54-61: 3-5, 8-14, 23, 27-29, 36, 38,
39, 41, 43, 45; pp. 61-64: 4-7, 11, 18
Those with the 4th edition should try the following problems. Ross pp. 56-57: 4-7, 10, 12; p. 59:
3, 4; pp. 60-61: 1-6, 8, 9, 12, 15-17, 23, 24, 26, 27, 29, 31, 33
Problems:
1. Consider an experiment with a finite sample space containing n equally likely outcomes.
Thus, there are 2n different events defined on this sample space.
If you cannot solve the problem for general n, solve it (for 50% credit) for
the case n = 3 and Ω = {a,b,c} discussed in class. In (d), assume outcome
b occurs.
(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 shown 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?
2. Five basketball teams play in a round-robin tournament, that is, each team plays the other
four teams exactly once.
(a) What is the total number of games played in the tournament?
(b) In each game, one team wears a dark uniform while the other wears a light uniform. Is it
possible to arrange matters such that each team wears dark uniforms for two of its games
and light uniforms for the other two?
(c) No game ends in a tie; one team wins and the other loses. If n denotes your answer to part
(a), then there are 2n different results that might occur, and we assume that all of 2n results
are equally likely. With this assumption, find the probability that each team wins at least
one game and loses at least one game (that is, no team has 4-0 or 0-4 record in the
tournament.)
3. The experiment consists of picking a letter at random from the word CHATTANOOGA.
(a) Define a sample space with 11 equally likely outcomes. What is the probability that the
letter picked is a vowel?
(b) Another way of setting up a probability space is to take Ω = {A,C,G,H,N,O,T}. Are the
seven outcomes equally likely? Do any of the outcomes have the same probability as some
other outcomes?
Now consider picking three letters at random (choosing a subset of size 3) from the letters
in the word CHATTANOOGA.
(c) What is the probability that the letters chosen can be arranged to form one of the common
words CAT, HAT, OAT, TAN and ANT? (Doesn’t matter which word is formed)
(d) Repeat part (c) assuming that sampling with replacement is being used
(e) Both for sampling with replacement and for sampling without replacement, find the
probability that the letters, as they are picked, form one of the 5 words of part (c)
(doesn’t matter which one) without having to be re-arranged.
4. An experiment consists of tossing a fair coin ten times
(a) State TRUE or FALSE: The sample space Ω comprises 10 equally likely outcomes.
(b) What is the probability that the third toss results in Head? Call this event A
(c) What is the probability that Head turns up exactly 5 times? Call this event B
(d) What is P(A ∪ B) ? What is P(A ∩ B)? What is the probability that exactly one of the
events A and B occurs, i.e. what is P(A ⊕ B)?
5.The manufacturer of a cereal tests samples from the production line to see if the samples
snap, crackle, and pop as advertised. Let A, B, and C denote respectively the events that
the sample does not snap, does not crackle, and does not pop. The manufacturer's
