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MA 421 Test I

# MA 421 Test I - b at least 4 of the problems 4 Let P be a...

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MA 421 - Exam I September 26, 2007 (Total = 100 points) Show all work! 1. Let A , B , and C be arbitrary subsets of a sample space S with probability measure P . Answer each of the following True or False (5 pts. each): a) ( A B ) C c = A ( B C c ) b) P ( A B ) P ( A ) + P ( B ) - 1 c) If P ( A ) > 0 and P ( B | A ) > P ( B ), then P ( B c | A ) < P ( B c ). 2. Let Δ denote the “exclusive or” operation on sets, that is, A Δ B = ( A B c ) ( B A c ) . a) (10 pts.) Express P ( A Δ B ) in terms of of P ( A ), P ( B ), and P ( A B ). b) (5 pts.) Is Δ an associative operation, that is, for sets A , B , and C , is it true that ( A Δ B C = A Δ( B Δ C )? (Yes or No). 3. An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 (and only 7) of the questions, what is the probability that he or she will answer correctly (10 pts. each): a) all 5 problems;
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Unformatted text preview: b) at least 4 of the problems. 4. Let P be a probability measure on a sample space S . For events A , B , and C , P ( A ) = . 7, P ( B ) = . 2, P ( C ) = . 5, P ( A ∩ C ) = . 3, A and B are independent, while B ∩ C = ∅ . (10 pts. each): a) What is the probability that none of the events A , B , C occur? b) What is the probability that exactly one of the events A , B , C occur? c) What is the probability that at most one of the events A , B , C occur? 5. (20 pts.) 80% of the members of a certain club are right handed. 20% of the righthanded members play tennis but only 10% of the lefthanded mem-bers play. If a tennis player is randomly chosen, what is the (conditional) probability the player is left handed?...
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