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Unformatted text preview: ection 7B, Independence 02/02/2010 1 2 Formally, we define two events A and B to be independent if any of the following is true: (Stochastically) Independent Events Intuitively, two events A and B are independent events if knowledge of the occurrence of one of them does not affect the likelihood that the other occurs. Two events which are not independent are called dependent . > > = = = ) ( in ) ( ); ( in ) ( : provided theorem, a by , definition the to equivalent ) ( ) ( ) ( ) ( ) ( ) ( independence of definition the ) ( ) ( ) ( ) ( c A P b B P B P A B P c A P B A P b B P A P B A P a 3 Example . A bowl contains 7 blue chips and 3 red chips . Two chips are drawn at random, in order and with replacement . Let S = set of all ordered pairs of distinct chips, A be the event that the first chip drawn is red , and B be the event that the second chip drawn is blue . Are the events A and B independent? Solution 1 : A B = event first is red and second is blue . 3 10 = 30 3 7 = 21 10 10 = 100 | S | = | A | = | B | = | A B | = 10 7 = 70 P ( A ) = P ( B ) = P ( A B) = P ( A ) P ( B ) = P ( A ) = 30/100 = 0.3 P ( B ) = 70/100 = 0.7 P ( A B) = 21/100 = 0.21 P ( A ) P ( B ) = (0.3)( 0.7) = 0.21 P ( A ) = 30/100 = 0.3 P ( B ) = 70/100 = 0.7 P ( A B) = 21/100 = 0.21 P ( A ) P ( B ) = P ( A ) = 30/100 = 0.3 P ( B...
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