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Unformatted text preview: would not consider them to
be physically independent. However, A = {2, 4, 6}, B = {3, 6}, and AB = {6}. So P (A) = 1 ,
2
P (B ) = 1 and P (AB ) = 1 . Therefore, P (AB ) = P (A)P (B ), which means that A and B are
3
6
mutually independent. This is a simple example showing that events can be mutually independent,
even if they are not physically independent.
Here is a ﬁnal note about independence of two events. Suppose A is independent of B. Then
P (Ac B ) = P (B ) − P (AB ) = (1 − P (A))P (B ) = P (Ac )P (B ),
so Ac is independent of B. Similarly, A is independent of B c , and therefore, by the same reasoning,
Ac is independent of B c . In summary, the following four conditions are equivalent: A is independent
of B , Ac is independent of B , A is independent of B c , Ac is independent of B c .
Let us now consider independence conditions for three events. The following deﬁnition simply
requires any one of the events to be independent of any one of the other events.
Deﬁnition 2.4.4 Events A, B, and C are pairwise independent...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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