# Therefore a is independent of b if and only if b is

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Unformatted text preview: valent to the statement that, as a function of the argument B for A ﬁxed, the conditional probability P (B |A) has all the properties of an unconditional probability measure P. Compare these properties to the probability axioms in Section 1.2. Intuitively, if one assumes a probability distribution P is given, and then later learns that an event A is true, the conditional probabilities P (B |A) as B varies, is a new probability distribution, giving a new view of the experiment modeled by the probability space. 2.4 2.4.1 Independence and the binomial distribution Mutually independent events Let A and B be two events for some probability space. Consider ﬁrst the case that P (A) &gt; 0. As seen in the previous section, it can be that P (B |A) = P (B ), which intuitively means that knowledge that A is true does not aﬀect the probability that B is true. It is then natural to consider the events to be independent. If P (B |A) = P (B ), then knowledge that A is true does aﬀect the probability that B is true, and it is natural to consider the events to be dependent ( (i.e. not independent). Since, by deﬁnition, P (B |A) = PP AB ) , the conditi...
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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.

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