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1.2. DISCRETE PROBABILITY DISTRIBUTIONS
23
Suppose next that
A
and
B
are disjoint subsets of Ω. Then every element
ω
of
A
∪
B
lies either in
A
and not in
B
or in
B
and not in
A
. It follows that
P
(
A
∪
B
)
=
∑
ω
∈
A
∪
B
m
(
ω
) =
∑
ω
∈
A
m
(
ω
) +
∑
ω
∈
B
m
(
ω
)
=
P
(
A
) +
P
(
B
)
,
and Property 4 is proved.
Finally, to prove Property 5, consider the disjoint union
Ω =
A
∪
˜
A .
Since
P
(Ω) = 1, the property of disjoint additivity (Property 4) implies that
1 =
P
(
A
) +
P
(
˜
A
)
,
whence
P
(
˜
A
) = 1

P
(
A
).
±
It is important to realize that Property 4 in Theorem 1.1 can be extended to
more than two sets. The general ±nite additivity property is given by the following
theorem.
Theorem 1.2
If
A
1
, . . . ,
A
n
are pairwise disjoint subsets of Ω (i.e., no two of the
A
i
’s have an element in common), then
P
(
A
1
∪ ··· ∪
A
n
) =
n
±
i
=1
P
(
A
i
)
.
Proof.
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 Spring '09
 scf

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