180asolns1 - 180A HW 1 Solutions October 4 2006 1 A coin is...

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180A HW 1 Solutions October 4, 2006 1) A coin is tossed four times resulting in a sequence of heads and tails. Let A: exactly two heads B: heads and tails alternate C: first two tosses are headed. (a) Which events, if any, are mutually exclusive? (b) Which events, if any, are subsets of other sets? Answer a) Events B and C are mutually exclusive. b) B A . 2) Compute P ( A c B c ) and P ( A c ( A B )) in terms of P ( A ) , P ( B ) , and P ( A B ) . Answer a) P ( A C B C ) = 1 - P ( A B ) (by DeMorgan) b) P ( A C ( A B )) = P (( A C A ) ( A C B )) = P ( A C B ) = P ( B ) - P ( A B ) . 3) Two dice are thrown. Let A: the sum is odd B: at least one die lands on an even number C: the sum is 7 Describe A B , A B , B C , A B c and A B C c and compute the corresponding probabilities. Answer We first note that A B , so A B = A and A B = B . Thus P ( A B ) = P ( A ) = 1 / 2 and P ( A B ) = P ( B ) = 3 / 4 . (Note that this can be computed by simply enumerating possibilities and noting that 27/36 of the possible die rolls include at least one even number.) We note that
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