Chapter 2 lecture notes
Math 431, Spring 2011
Instructor: David F. Anderson
Chapter 2: Axioms of Probability
Section 2.2: Sample Space and Events
We consider an experiment whose outcome is not predictable with certainty. However,
we suppose that the set of all possible outcomes is known.
Deﬁnition 1.
The set of all possible outcomes of an experiment is the sample space
of an
experiment, and is denoted by
S
. Subsets of
S
are called events
. Elements of
S
,
x
∈
S
, are
called
outcomes
.
•
In this course, we will care about the probability of both events and speciﬁc outcomes.
That is,
P
(
E
) and
P
(
{
x
}
) =
P
(
x
).
Example 1.
A coin is tossed twice and the outcome of each is recorded. Then,
S
=
{
(
H,H
)
,
(
H,T
)
,
(
T,H
)
,
(
T,T
)
}
.
The event that the second toss was a Head is the subset
E
=
{
(
H,H
)
,
(
T,H
)
}
.
Example 2.
Consider 3 lightbulbs. Our experiment consists of ﬁnding out which lightbulb
burns out ﬁrst, and how long (in hours) it takes for this to happen.
S
=
{
(
i,t
) :
i
∈ {
1
,
2
,
3
}
,t
≥
0
}
,
i
tells you which one burns out, and
t
gives how long it lasted, in hours. The event
that the
2nd bulb burns out ﬁrst, and it lasts less than 3 hours is the set
E
=
{
(2
,t
) :
t <
3
}
.
Example 3.
You roll a four sided die until a 4 comes up. The event you are interested in
is getting a three with the ﬁrst two rolls.
S
=
{
(
a
1
,a
2
,
···
,a
n
)

n
≥
1
,a
n
= 4
,a
i
∈ {
1
,
2
,
3
}
for
i
6
=
n
}
E
=
{
(3
,
3
,a
3
,a
4
,
···
,a
n
)

n
≥
3
,a
n
= 4
,a
i
∈ {
1
,
2
,
3
}
for
i
6
=
n
}
1