STAT 230: Introduction to Probability and
Random Variables
Summer 2009
1
Probability
1.1
Sample space and Events
Definition 1.
Random Experiment
A random experiment is an experiment for which the outcomes cannot be deter-
mined ahead of time.
Definition 2.
Sample Space
The sample space
Ω
is the collection of all possible outcomes. Elements
ω
in
Ω
are called simple outcomes.
Definition 3.
Event
An event is a subset of the sample space. We say that an event occurs if and
only if the outcome of the random experiment is an element of the event.
Example 1.
For example, consider the experiment where you draw a random
card from a standard deck. The sample space consists of the 52 cards and one
event is A =
{
card drawn is red
}
.
Example 2.
Suppose the experiment consists of tossing a coin twice.
The
sample space is
Ω =
{
(
H, H
)
,
(
H, T
)
,
(
T, H
)
,
(
T, T
)
}
. The outcomes in the event
A =
{
first toss is Head
}
are
{
(H,H),(H,T)
}
.
Example 3.
Consider the experiment where a pair of 6-sided die are rolled.
The sample space is
Ω = (
i, j
);
i
= 1
..
6
, j
= 1
..
6
.
The outcomes in A =
{
the
second roll is 3
}
are
{
(i,3);i=1..6
}
.
Example 4.
Suppose we roll a 6-sided die and toss a coin. For the events A =
{
the die is even
}
and B =
{
the coin is Head
}
, then:
A =
{
(2,H),(2,T),(4,H),(4,T),(6,H),(6,T)
}
.
B =
{
(1,H),(2,H),(3,H),(4,H),(5,H),(6,H)
}
.
A’ =
{
(1,H),(1,T),(3,H),(3,T),(5,H),(5,T)
}
.
B’ =
{
(1,T),(2,T),(3,T),(4,T),(5,T),(6,T)
}
.
A
∩
B
=
{
(2,H),(4,H),(6,H)
}
.
1