Math170a.test1.solutions170a

# Solution 1 n k b more generally what is probability

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Solution: 1 / ( n k ) . (b) More generally, what is probability that L contains exactly j of your choices? Solution: ( k j )( n - k k - j ) / ( n k ) . (10) 4. Two fair dice are rolled. Consider the following events: A, that the first die shows a 1. B, that the sum of the two outcomes is 7. C, that the sum of the two outcomes is 4. (a) Are A and B independent? Explain. Solution: P ( A ) = 1 6 ,P ( B ) = 1 6 ,P ( A B ) = 1 36 , so P ( A B ) = P ( A ) P ( B ), and they are independent. (b) Are A and C independent? Explain. Solution: P ( A ) = 1 6 ,P ( C ) = 1 12 ,P ( A C ) = 1 36 , so P ( A B ) negationslash = P ( A ) P ( B ), and they are not independent. (20) 5. A fair coin is tossed until the first head occurs. Let N be the number of the toss at which that happens. Then N fair dice are tossed. Let S be the sum of the results of the N dice. (a) Find P ( N = 5). Solution: P ( N = 5) = 1 2 5 . (b) Find P ( S = 3). Solution: P ( S = 3) = P ( S = 3 | N = 1) P ( N = 1) + P ( S = 3 | N = 2) P ( N = 2) + P ( S = 3 | N = 3) P ( N = 3) = 1 6 1 2 + 1 18 1 4 + 1 216 1 8 = 169 1728 .

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(c) Find P ( N = 2 | S = 3). Solution: P ( N = 2 | S = 3) = P ( S = 3 | N = 2) P ( N = 2) /P ( S = 3) = 1 72 / 169 1728 = 24 169 . (20) 6. A hand of 13 cards is drawn from a standard 52 card deck. (a) Find the probability that the hand consists of 4 spades, 3 hearts, 3 diamonds and 3 clubs. Solution: ( 13 4 )( 13 3 ) 3 / ( 52 13 ) . (b) Find the probability that the hand consists of four cards of one suit and three cards of each of the other three suits. Solution: 4 ( 13 4 )( 13 3 ) 3 / ( 52 13 ) .
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