Generally, there are three interpretations of probability:
1
Classical probability;
2
Empirical or relative frequency probability; and
3
Subjective probability
Classical interpretation of probability arose from games of chance. It uses
sample space to determine numerical probability that an event will happen.
The assumption is that all outcomes in the sample space are equally likely
to occur.
Don’t just assume that all outcomes are equally likely unless either
you are told to, or there is some physical reason for assuming it!
By Arthur Mpazi Yambayamba
Introduction to Probability and Probability Distributions
September 1, 2015
5 / 20

Classical Probability
If there are
n
equally likely possibilities, of which one must occur, and
m
of
these are regarded as favourable to an event, or as “success”, then the
probability of the event or a “success”is given by
m
n
. When all outcomes
are equally likely:
1
count the number of outcomes in the sample space; say this is
n
.
2
count the number of outcomes in the event of interest,
A
, and say
this is
m
.
3
P
(
A
) =
m
n
.
Example 1
: A balanced die (with all outcomes equally likely) is rolled. Let
A
be the event that an even number occurs. There are six elements in the
sample space,
S
=
{
1
,
2
,
3
,
4
,
5
,
6
}
, and there are three favourable
outcomes (2, 4, 6) in
A
. Hence,
P
(
A
) =
3
6
=
1
2
.
By Arthur Mpazi Yambayamba
Introduction to Probability and Probability Distributions
September 1, 2015
6 / 20

Classical Probability...Cont’d
Example 2
: Suppose we toss two coins. Assume that the outcomes are
equally likely (fair coin).
(a)
What is the sample space?
(b)
Let
A
be the event that at least one of the coins shows up heads. Find
P
(
A
).
(c)
What will be the sample space if we know that at least one of the
coins showed up heads?
(a)
The sample space consists of four outcomes, namely
S
=
{
(
H
,
H
)
,
(
H
,
T
)
,
(
T
,
H
)
,
(
T
,
T
)
}
.
(b)
The event
A
has three outcomes, (H, H), (H, T), and (T, H).
Therefore,
P
(
A
) =
3
4
.
(c)
Since we know that at least one of the coins showed up heads, the
possible outcomes are (H, H), (H, T), and (T, H). The sample space
now has only three outcomes,
S
=
{
(
H
,
H
)
,
(
H
,
T
)
,
(
T
,
H
)
}
.
By Arthur Mpazi Yambayamba
Introduction to Probability and Probability Distributions
September 1, 2015
7 / 20

Empirical or Relative Frequency Probability
The
probability
of an outcome (event) is the proportion of times the
outcome (event) would occur in a long run of repeated experiments.
Empirical probability relies on actual experience to determine the likelihood
of outcomes. In empirical probability, for example, one might actually roll
a given die or toss a coin 5,000 times, observe the various frequencies, and
use these frequencies to determine the probability of an outcome!
Symbolically, if an experiment is conducted
n
different times and if
event
A
occurs on
n
A
of these trials, then the probability of event
A
is
approximately:
P
(
A
)
≈
n
A
n
.

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