midterm1-2 - Math 310-1 Exam 1 10.27.2009 Directions: Write...

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Math 310-1 Exam 1 10.27.2009 Directions: Write your name and your TA’s name on the cover of your exam booklet. Show all work and clearly mark your final answers. Answers must be sim- plified unless otherwise indicated (by a footnote). You may use one sheet of paper with anything you want written on it, but you may not use a calculator. 1. (10 pts) Let A, B and C be events, each having probability 1 / 4. Suppose A and B are independent, A and C are disjoint, and P ( C | B ) = 1 / 3. Find P ( A B C ). 2. You have a fair die and a coin that flips heads 2 / 3 of the time. Suppose you roll the die and then flip the coin the number of times that the die shows (i.e. if you roll a 3, you flip the coin three times). (a) (10 pts) What is the probability that you flip exactly four heads? 1 (b) (10 pts) Show that the probability that you rolled a five, given that you flipped exactly four heads, is 5 / 13. 3. Let X be exponential with parameter λ . (a) (10 pts) What is the probability X is less than 2 or greater than 4? (b) (10 pts) Let Y = 1 / ( X + 1). Find the distribution function of Y . (c) (10 pts) Let Z = X . Find a density function of Z . 4. Suppose X is geometric with parameter p . (a) (10 pts) Let 0 a b . Find P ( X b | X a ). (b) (10 pts) Let Y = ± X if X 3 4 if X 4 . Find the density function of Y . 5. Suppose you deal a six-card hand from a standard 52-card deck. (a) (6 pts) What is the probability that your six-card hand is a flush (i.e. your cards all belong to the same suit)? 1 (b) (6 pts) What is the probability your hand contains 3 of a kind and a pair (i.e. your six cards have values x , x , x , y , y , and z , where x, y, z are different)? 1
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midterm1-2 - Math 310-1 Exam 1 10.27.2009 Directions: Write...

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