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HW7_sol - CSE21 FA11 Homework#7 Solutions 7.1 A bin...

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CSE21 FA11 Homework #7 Solutions 7.1. A bin contains 4 red balls, 5 white balls and 6 blue balls. A random subset S of 4 balls is removed (without replacement). Consider the following 3 events: (1) E 1 : S has exactly 2 red balls.; (2) E 2 : S has balls of all three colors; (3) E 3 : S has at least 2 blue balls. Which of the pairs of events E 1 , E 2 and E 3 (if any) are independent? (Justify your answer.) Answer: None of the pairs are independent. For example, take the pair E 1 and E 2 . We have to compute Pr ( E 1 ) , P ( E 2 ) and Pr ( E 1 T E 2 ). Since the sample space has size ( 15 4 ) and all these choices are equally likely, then: Pr ( E 1 ) = ( 4 2 )( 11 2 ) ( 15 4 ) , Pr ( E 2 ) = ( 4 2 )( 5 1 )( 6 1 ) + ( 4 1 )( 5 2 )( 6 1 ) + ( 4 1 )( 5 1 )( 6 2 ) ( 15 4 ) and Pr ( E 1 T E 2 ) = ( 4 2 )( 5 1 )( 6 1 ) ( 15 4 ) . Now we have to check if Pr ( E 1 T E 2 ) = Pr ( E 1 ) Pr ( E 2 ). Since this doesn’t hold, then the events E 1 and E 2 are not independent. Similar calculations for the other two pairs show that they are also not independent. 7.2. We are given an urn that has one red ball and two white balls in it. A fair die is thrown. If the number shown is 1 or 2 then one red ball is added to the urn. Otherwise three red balls are added to the urn. A ball is then randomly drawn from the urn.
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