midterm1-solutions

midterm1-solutions - Math 310-1 Midterm 1 Fall 2011...

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Unformatted text preview: Math 310-1 Midterm 1 Fall 2011 Instructions: Do all work and record all answers in the blue book. No books, notes, calculators are allowed. You must show all of your work. DO NOT simplify factorials, binomial coefficients or multinomial coefficients, unless asked to do so. 1. (18 points): A fair coin is tossed twice. Let A = “both head and tail appear”, B = “first toss is tail”, and C = “second toss is tail”. (a) Are A , B , C pairwise independent? (b) Are A , B , C mutually independent? Solution: (a) The sample space is Ω = { HH,HT,TH,TT } . So, P ( A ) = P ( { HT,TH } ) = 2 / 4 = 1 / 2 P ( B ) = P ( { TH,TT } ) = 2 / 4 = 1 / 2 P ( C ) = P ( { HT,TT } ) = 2 / 4 = 1 / 2 Now, P ( A ∩ B ) = P ( { TH } ) = 1 / 4 = P ( A ) P ( B ) = ⇒ A and B are independent P ( A ∩ C ) = P ( { HT } ) = 1 / 4 = P ( A ) P ( C ) = ⇒ A and C are independent P ( B ∩ C ) = P ( { TT } ) = 1 / 4 = P ( B ) P ( C ) = ⇒ B and C are independent. Hence, the events A , B , and C are pairwise independent. (b) Notice that A ∩ B ∩ C = ∅ . So, P ( A ∩ B ∩ C ) = 0 Ó = 1 / 8 = P ( A ) P ( B ) P ( C ) . Therefore, the events A , B , and C are not mutually independent. Page 1 of 6 Math 310-1 Midterm 1 Fall 2011 2. (30 points): Urn I contains 7 blue, 5 green and 6 yellow balls; urn II contains 4 blue, 6 green and 2 yellow balls; urn III contains 2 blue, 4 green and 3 yellow balls. One of the urns is chosen at random and then a ball is randomly selected from the chosen urn. (a) What is the probability that the chosen ball is yellow? (b) Suppose that a yellow ball has been chosen. What is the probability that urn I was picked?...
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midterm1-solutions - Math 310-1 Midterm 1 Fall 2011...

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