University of Illinois
Fall 2010
ECE 313:
Problem Set 1: Solutions
Axioms of probability and calculating the sizes of sets
1.
[Deﬁning a set of outcomes]
(a) One choice would be Ω =
{
(
w
1
,w
2
3
) :
w
1
∈ {
1
,
2
}
2
∈ {
3
,
4
}
3
∈ {
w
1
2
}}
,
where
w
i
denotes which team wins the
i
th
game, for 1
≤
i
≤
3
.
Another choice would be Ω =
{
(
x
1
,x
2
3
) :
x
i
∈ {
L,H
}
for 1
≤
i
≤
3
}
,
where
x
i
indicates which
of the two teams playing the
i
th
game wins;
x
i
=
H
indicates that the higher numbered team
wins and
x
i
=
L
indicates that the lower numbered team wins. For example, (
L,L,H
) indicates
that team one wins the ﬁrst game, team three wins the second game (so team one plays team
three in the third game), and team three wins the third game.
(b) Eight, because there are two possible outcomes for each of the three games, and 2
3
= 8
.
2.
[Possible probability assignments]
(This is one of many ways to get the answer.) Since, by De Morgan’s law, the complement of
A
∪
B
is
A
c
B
c
, the fact
P
(
A
∪
B
) = 0
.
6 is equivalent to
P
(
A
c
B
c
) = 0
.
4
.
Thus, we can ﬁll in the Karnaugh
diagram for
A
and
B
as shown:
0.3
B
c
A
A
c
B
0.3
0.4
a
!
a
We ﬁlled in the variable
a
for
P
(
AB
c
)
,
and then, since the sum of the probabilities is one, it must be
that
P
(
A
c
B
) = 0
.
3

a.
The valid values of
a
are 0
≤
a
≤
0
.
3
,
and (
P
(
A
)
,P
(
B
)) = (0
.
3 +
a,
0
.
6

a
)
.
So, in parametric form, the set of possible values of (
P
(
A
)
(
B
)) is
{
(0
.
3 +
a,
0
.
6

a
) : 0
≤
a
≤
0
.
3
}
.
Equivalent ways to write this set are
{
(
u,v
) :
v
= 0
.
9

u
and 0
.
3
≤
u
≤
0
.
6
}
or
{
(
x,
0
.
9

x
) : 0
.
3
≤
x
≤
0
.
6
}
.
The set is also represented by the solid line segment in the following sketch:
0.3
P(B)
P(A)
0.3
0.6
0.9
0.9
0.6
3.
[Grouping students into teams]
(a) One solution is the following. Five teams, numbered one through ﬁve, can be sequentially selected
as follows. To begin, there are
(
10
2
)
ways to choose team one. That leaves eight students, so for
any choice of team one, there are
(
8
2
)
ways to choose team two. That leaves
(