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2. Mathematical Probability Models
2.1
Sample Spaces and Probability
Consider some phenomenon or process which is repeatable, at least in theory, and suppose that certain
events or outcomes
A
1
,A
2
,A
3
,...
are defined. We will often term the phenomenon or process an
“experiment"
and refer to a single repetition of the experiment as a
“trial"
. The probability of an
event
A
, denoted
P
(
A
)
, is a number between 0 and 1. For probability to be a useful mathematical
concept, it should possess some other properties. For example, if our “experiment” consists of tossing
a coin with two sides, Head and Tail, then we might wish to consider the two events
A
1
=“Headturns
up” and
A
2
= “Tail turns up”. It does not make much sense to allow
P
(
A
1
)=0
.
6
and
P
(
A
2
)=0
.
6
,so
that
P
(
A
1
)+
P
(
A
2
)
>
1
. (Why is this so? Is there a fundamental reason or have we simply adopted
total probability=
1
as a convenient scale?) To avoid this sort of thing we begin by defining the list of
all possible outcomes and then assign probabilities adding to one.
Definition 1
A
sample space
S
is a set of distinct outcomes for an experiment or process, with the
property that in a single trial, one and only one of these outcomes occurs. The outcomes that make up
the sample space are called
sample points
.
A sample space is part of the probability model in a given setting. It is not necessarily uniquely defined,
as the following example shows.
Example:
Roll a 6sided die, and define the events
a
i
=
number
i
turns up,
(
i
=1
,
2
,
3
,
4
,
5
,
6)
Then we could take the sample space as
S
=
{
a
1
,a
2
,a
3
,a
4
,a
5
,a
6
}
. However, we could also define
events
E
=
even number turns up
O
=
odd number turns up
and take
S
=
{
E,O
}
. Both sample spaces satisfy the definition. Which one we use would depends on
what we wanted to use the probability model for. If we expect
never
to have to consider events like "a
5
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number less than 3 turns up" then the space
S
=
{
E,O
}
will suffice, but in most cases, if possible,
S
is
comprised of sample points that are the smallest possible or "indivisible". Thus the first sample space
is likely preferred in this example.
Sample spaces may be either
discrete
or
nondiscrete
;
S
is discrete if it consists of a finite or
countably infinite set of simple events. Recall that a countably infinite sequence is one that can be
put in oneone correspondence with the positive integers, so for example
{
1
2
,
1
3
,
1
4
,
1
5
,...
}
is countably
infinite as is the set of all rational numbers. The two sample spaces in the preceding example are
discrete. A sample space
S
=
{
1
,
2
,
3
,...
}
consisting of all the positive integers is discrete, but a
sample space
S
=
{
x
:
x>
0
}
consisting of all positive real numbers is not. For all but the last chapter
in these notes we consider only discrete sample spaces. This makes it easier to define mathematical
probability, as follows.
Definition 2
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This note was uploaded on 02/11/2010 for the course ART AFM101 taught by Professor Mr.lushman during the Spring '10 term at University of Toronto Toronto.
 Spring '10
 Mr.Lushman

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