# HW1 - chosen The chance of Anna being chosen is 40 Give...

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OR 3500/5500, Summer’09, Chen Homework 1 Due on Wednesday, May 20, 3:00 pm. For each problem just giving the answer will not suﬃce; a proper argument is required. Problem 1 A fair die is rolled twice. (a) List the elements in the following events: A =At least one of the rolls is 6. B =The sum of the rolls is 8. C =Product of the two rolls is divisible by 6. (b) Compute P ( A ), P ( B ) and P ( C ) using the classical deﬁnition of probability. Problem 2 Prove rigorously, using the axioms and derived properties of probability (a) If A B , then P ( A ) P ( B ) (b) Boole’s Inequality: for any (not necessarily disjoint) sets A 1 ,A 2 ,...A n , P ˆ n [ i =1 A i ! n X i =1 P ( A i ) . (c) Bonferroni’s inequality: for any sets A 1 ,A 2 ,...A n , P ˆ n \ i =1 A i ! n X i =1 P ( A i ) - ( n - 1) . Problem 3 Anna, Donna and Elena are sophomores now and it’s time for the selection of the women’s hockey team again. Of course the chances of selection have totally changed from what it was in their ﬁrst years(lecture notes). Anna and Elena are the only players who play as goal-keepers. Hence, exactly one of them has to be

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Unformatted text preview: chosen. The chance of Anna being chosen is 40%. Give answers and illustration using venn-diagrams for the following: (a) The chance of both Anna and Donna being chosen is 20%. What is the maximum and minimum chance of Donna being chosen? (b) If the chance of Donna being chosen is 70%, what is the maximum and minimum chance of Anna and Donna both being chosen? (c) If the chance of Donna being chosen is 50% and that of both Donna and Anna being chosen is 20%, what is the chance that both Donna and Elena are chosen? The following problem is optional for 3500, compulsory for stu-dents enrolled in 5500 1 OR 3500/5500, Summer’09, Chen Problem 4 A number is chosen at random from { 1 , 2 ,...,n } . Find the probability that the number is divisible by (a)2 (b)3 (c)either 2 or 3 (use inclusion-exclusion formula) Denoting the probability in (c) by p n , compute lim n →∞ p n 2...
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HW1 - chosen The chance of Anna being chosen is 40 Give...

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