CHAPTER
2
PROBABILITY
2.1
Sample Space
A
probability model
consists of the sample space and
the way to assign probabilities.
Sample space & sample point
The
sample space
S
, is the set of all possible outcomes
of a statistical experiment.
Each outcome in a sample space is called a
sample
point
. It is also called an
element
or a
member
of the
sample space.
For example, there are only two outcomes for
tossing a coin, and the sample space is
S
=
{
heads, tails
}
,
or
,
S
=
{
H, T
}
.
If we toss a coin three times, then the sample space is
S
=
{
HHH, HHT, HTH, THH, HTT, TTH, THT, TTT
}
.
E
XAMPLE
2.1.
Consider rolling a fair die twice and
observing the dots facing up on each roll. What is the
sample space?
There are 36 possible outcomes in the sample
space
S
, where
S
=
(
1
,
1
)
(
1
,
2
)
(
1
,
3
)
(
1
,
4
)
(
1
,
5
)
(
1
,
6
)
(
2
,
1
)
(
2
,
2
)
(
2
,
3
)
(
2
,
4
)
(
2
,
5
)
(
2
,
6
)
(
3
,
1
)
(
3
,
2
)
(
3
,
3
)
(
3
,
4
)
(
3
,
5
)
(
3
,
6
)
(
4
,
1
)
(
4
,
2
)
(
4
,
3
)
(
4
,
4
)
(
4
,
5
)
(
4
,
6
)
(
5
,
1
)
(
5
,
2
)
(
5
,
3
)
(
5
,
4
)
(
5
,
5
)
(
5
,
6
)
(
6
,
1
)
(
6
,
2
)
(
6
,
3
)
(
6
,
4
)
(
6
,
5
)
(
6
,
6
)
N
OTE
.
You may use a
tree diagram
to systematically
list the sample points of the sample space.
E
XAMPLE
2.2.
A fair sixsided die has 3 faces that are
painted blue (
B
), 2 faces that are red (
R
) and 1 face that
is green (
G
). We toss the die twice. List the complete
sample space of all possible outcomes.
(a) if we are interested in the
color
facing upward on
each of the two tosses.
(b) if the outcome of interest is the
number
of red we
observe on the two tosses.
N
OTE
.
A
statement
or
rule method
will best describe
a sample space with a large or infinity number of sample
points. For example, if
S
is the set of all points
(
x
,
y
)
on
the boundary or the interior of a unit circle, we write a
rule/statement
S
=
{
(
x
,
y
)

x
2
+
y
2
≤
1
}
.
E
XAMPLE
2.3.
List the elements of each of the follow
ing sample spaces:
(a)
S
=
{
x

x
2

3
x
+
2
=
0
}
(b)
S
=
{
x

e
x
<
0
}
N
OTE
.
The
null set
, or empty set, denoted by
φ
, con
tains no members/elements at all.
2.2
Events
Event
An
event
is a subset of a sample space.
Refer to Example
2.1
.
“the sum of the dots is
6” is an event. It is expressible of a set of elements
E
=
{
(
1
,
5
)
(
2
,
4
)
(
3
,
3
)
(
4
,
2
)
(
5
,
1
)
}
Complement
The event that
A
does not occur, denoted as
A
0
, is
called the
complement
of event
A
.
E
XAMPLE
2.4.
Refer to Example
2.1
.
What are the
complement events of