Homework2Solution - Homework 2 Solutions p111...

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Homework 2 Solutions July 23, 2007 p111 10,13,28,59,76,79 p125 7a, 12 1. #10 Three cards are selected without replacement from an ordinary deck of 52 playing cards. Compute the conditional probability that the first card selected is a spade given that the second and third cards are spades. Denote the i th card drawn by C i . P ( C 1 = s | C 2 = s, C 3 = s ) = P ( C 1 = s,C 2 = s,C 3 = s ) P ( C 2 = s,C 3 = s ) . P ( C 1 = s, C 2 = s, C 3 = s ) = ( 13 3 ) ( 52 3 ) . P ( C 2 = s, C 3 = s ) = ( 13 2 ) ( 52 2 ) . We get 13!3!49! 3!10!52! * 13!2!50! 2!11!52! = 11 50 . 2. #13 Suppose that an ordinary deck of 52 cards is randomly divided into 4 hands of 13 cards each. Determine p , the probability that each hand has an ace. Let E i be the event that the i th hand has exactly one ace. Determine p = P ( E 1 E 2 E 3 E 4 ) using the multiplication rule. By the multiplication rule, P ( E 1 E 2 E 3 E 4 ) = P ( E 4 | E 1 E 2 E 3 ) * P ( E 3 | E 2 E 1 ) * P ( E 2 | E 1 ) * P ( E 1 ). P ( E 1 ) = ( 4 1 )( 48 12 ) ( 52 13 ) . P ( E 2 | E 1 ) = ( 3 1 )( 36 12 ) ( 39 13 ) . P ( E 3 | E 1 E 2 ) = ( 2 1 )( 24 12 ) ( 26 13 ) . 1
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P ( E 4 | E 1 E 2 E 3 ) = 1. Then P ( E 1 E 2 E 3 E 4 ) = ( 4 1 )( 48 12 ) ( 52 13 ) * ( 3 1 )( 36 12 ) ( 39 13 ) * ( 2 1 )( 24 12 ) ( 26 13 ) . 3. #28 Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional probability that the card following it is the (a) ace of spades P ( C 21 =
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