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55010Note3

# 55010Note3 - J Chen Handout 3 STAT550 2010 1 STAT550...

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J. Chen, Handout 3, STAT550, 2010 1 STAT550 Applied Probability http://www-rohan.sdsu.edu/ jchenyp/STAT5502010.htm 1.5 Conditional Probability We want to ﬁnd the conditional probability of an event A given that some other event B has occurred. The information that an event B has occurred may aﬀect the probability of event A . The symbol P ( A | B ) read ”the probability of A given B ” - is used to denote a conditional probability. P ( A | B ) refers to the PROBABILITY that A will occur given that B has already occurred. EX1. Consider rolling a die. If we know that the outcome of a roll of a die is even, what is the probability of the outcome 1, 6? We deﬁne A is ”1, 6 appear” and B is ”even number appear ”. Clearly, P ( A ) = 2 6 , P ( B ) = 3 6 , P ( A B ) = 1 / 6 and P ( A | B ) = 1 3 = 1 / 6 3 / 6 = P ( A B ) P ( B ) Therefore the conditional probability of A given B is the ratio of P ( A B ) and P ( B ). Deﬁnition 1.5-1. Let A and B be any two events deﬁned on a sample space S such that P ( B ) > 0. Then the conditional probability of A given B is given by P ( A | B ) = P ( A B ) P ( B ) . The deﬁnition of P ( A | B ) yields P ( A B ) = P ( A | B ) P ( B ) Call Multiplication Rule. Also we have P ( A B ) = P ( B | A ) P ( A ) Both rules allow us to use the conditional probability to evaluate the intersection probability. Notice that the following properties will be useful.

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J. Chen, Handout 3, STAT550, 2010 2 1) Based on the deﬁnition of the conditional probability, it follows that P ( A | B C ) = P ( A B C ) P ( B C ) , P ( A | B C ) = P ( A B C ) P ( B C ) , P ( A | B C ) = P ( A ( B C )) P ( B C ) . 2) Specially, when B is the sample space, then P ( A | S ) = P ( A S ) P ( S ) = P ( A ) So the probability of A is a special conditional probability. 3) The conditional probability P ( A | B ) satisﬁes the three Axiom properties, i.e. i) 0 < P ( A | B ) < 1; ii) P ( B | B ) = 1, and iii) P ( A 1 A 2 | B ) = P ( A 1 | B )+ P ( A 2 | B ), if A 1 and A 2 are disjoint. Similarly, the result holds for higher-order intersections. Consider three events A , B , and C . The conditional probability P ( A B C ) can be written as P ( A B C ) = P ( A ) P ( B | A ) P ( C | A B ) = P ( B ) P ( A | B ) P ( C | A B ) = P ( C ) P ( A | C ) P ( B | A C ) In general, for n events A 1 ,A 2 , ··· ,A n , we have the the following formula P ( A 1 A 2 ∩ ··· ∩ A n ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 2 ) ··· P ( A n - 1 | A 1 A 2 ∩ ··· A n - 2 ) × P ( A n | A 1 A 2 ∩ ··· A n - 1 ) EX2. A card is drawn from a poker deck. What is the probability that the card is a club, given that the card is a king? Solution:
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55010Note3 - J Chen Handout 3 STAT550 2010 1 STAT550...

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