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. These are the probabilities that the input is 0 and the output is 1,
and the input is 1 and the output is 1. The event
C
—that is, the output is 1—will occur only
when one of the two
joint
events occurs, therefore,
P
(
C
|
B
)
,
P
(
A
)
, and
P
(
B
)
P
(
C
)
=
P
(
A
,
C
)
+
P
(
B
,
C
)
Similarly,
P
(
D
)
=
P
(
A
,
D
)
+
P
(
B
,
D
)
Numerically, this comes out to be
P
(
C
)
=
P
(
D
)
=
0
.
5
A.1.3 The Axiomatic Approach
Finally, there is an approach that simply defines probability as a measure, without much regard
for physical interpretation. We are very familiar with measures in our daily lives. We talk
about getting a 9-foot cable or a pound of cheese. Just as length and width measure the extent
of certain physical quantities, probability measures the extent of an abstract quantity, a set. The
thing that probability measures is the “size” of the event set. The probability measure follows
similar rules to those followed by other measures. Just as the length of a physical object is
always greater than or equal to zero, the probability of an event is always greater than or equal
to zero. If we measure the length of two objects that have no overlap, then the combined length
of the two objects is simply the sum of the lengths of the individual objects. In a similar manner
the probability of the union of two events that do not have any outcomes in common is simply
the sum of the probability of the individual events. So as to keep this definition of probability
in line with the other definitions, we normalize this quantity by assigning the largest set, which
is the sample space
S
, the size of 1. Thus, the probability of an event always lies between 0
and 1. Formally, we can write these rules down as the three
axioms
of probability.
Given a sample space
S
:
Axiom 1:
If
A
is an event in
S
, then
P
(
A
)
0.
Axiom 2:
The probability of the sample space is 1; that is,
P
(
S
)
=
1.
Axiom 3:
If
A
and
B
are two events in
S
and
A
∩
B
=
φ
, then
P
(
A
∪
B
)
=
P
(
A
)
+
P
(
B
)
.
Given these three axioms we can come up with all the other rules we need. For example,
suppose
A
c
is the complement of
A
. What is the probability of
A
c
? We can get the answer by
using Axiom 2 and Axiom 3. We know that
A
c
∪
A
=
S
and Axiom 2 tells us that
P
(
S
)
=
1, therefore,
A
c
P
(
∪
A
)
=
1
(A.5)
