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Unformatted text preview: on P (B |A) = P (B ) is
equivalent to P (AB ) = P (A)P (B ).
Let’s consider the other case: P (A) = 0. Should we consider A and B to be independent? It
doesn’t make sense to condition on A, but P (Ac ) = 1, so we can consider P (B |Ac ) instead. It holds
that P (B |Ac ) = PP A c )) = P (Ac B ) = P (B ) − P (AB ) = P (B ). Therefore, P (B |Ac ) = P (B ). That
is, if P (A) = 0, knowledge that A is not true does not aﬀect the probability of B. So it is natural
to consider A to be independent of B.
These observations motivate the following deﬁnition, which has the advantage of applying
whether or not P (A) = 0 :
Deﬁnition 2.4.1 Event A is independent of event B if P (AB ) = P (A)P (B ).
Note that the condition in the deﬁnition of independence is symmetric in A and B. Therefore,
A is independent of B if and only if B is independent of A. Another commonly used terminology for
these two equivalent relations is to say that A and B are mutually independent. Here, “mutually”
means that independence is a property of the tw...
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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 Tech.
- Spring '08
- The Land