Feb 21’10
GENERAL ADDITION AND MULTIPLICATION RULES,
#5
CONDITIONAL
PROBABILITY, AND
BAYES’ THEOREM
General Addition (Union) Rule
.
For two given events,
A
and
B
, the probability that
event
A
, or event
B
, or both occur is equal to the probability that event
A
occurs, plus the
probability that event
B
occurs,
minus
the probability that the events occur together. This
addition rule may be written as
P(A
U
B)
=
P(A)
+
P(B) – P(A
∩
B)
(5.1)
For
mutually exclusive
events
A
and
B
the probability
P(A
and
B)
=
P(A
∩
B)
= 0
and the
rule (5.1)
has a simpler form
P(A
U
B) = P(A) + P(B)
(see formula (
4
.5) in handout #
4
).
/*
The formula (
4
.5) can be expanded.
In
the
case of
k
mutually
exclusive
events
E
1
,
E
2
, …,
E
k
the probability of occurrence of
any
of them (we treat it as an event
E
) is
k
P(E) =
P
(
E
1
U
E
2
U
E
3
U …U
E
k
) =
Σ
P(E
i
)
(5.2)
i =1
The relation (5.2) is known as the
theorem of addition of probabilities
. An event
E
defined in accordance with (5.2) is called
the sum of events
E
1
+
E
2
+…+
E
k
. In a special
case, when the events
E
1
,
E
2
, …,
E
k
constitute a
whole
sample space for some activity,
the sum (5.2) is equal to
1
due to the
property of exhaustiveness
(
4
.4).
*/
We should apply the general addition rule (5.1) instead of (
4
.5) when the events
A
and
B
under consideration are
not
mutually exclusive. In such a case, two sample spaces
S
A
and
S
B
associated with these events
intersect
. In other words, two sets of sample points
(representing the outcomes of an experiment) related to the events
A
and
B
have at least
one common sample point.
Let us illustrate the idea with the following thought
experiment_1
: if we roll a
single
die then the event
A
of showing an
odd number
and the event
B
of showing a
number
greater than
3 are
not
mutually exclusive. Such an event
A
can be realized in three ways:
{1}, {3}, {5};
in other words, the sample space
S
A
= {1, 3,
5
}
. Similarly, the sample space
S
B
=
{4,
5
, 6}
. We see that two sample spaces
S
A
and
S
B
intersect, and the outcome
{
5
}
is
the common sample point.
The situation may be illustrated also graphically, by means of a corresponding
Venn
diagram
.
Venn diagrams are simple “area” or “region” diagrams which are used to
illustrate the relationship between different events
. Usually they look like two or more
circles, each including a set of sample points (sample space) related to one of the events
under consideration. If the events do not have any point in common then the circles do
not intersect; that is, the events under consideration are mutually exclusive. But if the