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Practice_Final

# Practice_Final - Name Student ID CSE 21 Practice Final Exam...

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Name: Student ID: CSE 21 Practice Final Exam December 2, 2009 1. Define the recurrence a ( n + 2) = 4 a ( n + 1) - a ( n ) , , n 0, with a (0) = 1 , a (1) = 2. (a) What is the value of a (5)? (b) Find an explicit closed form solution for a ( n ); (c) Prove by induction that the expression in (b) is valid for all n . 2. Let x ( n + 1) = 2 x ( n ) + n, n 1, with x (1) = 1. Find a closed form expression for x ( n ). 3. An urn contains 3 Red, 4 White and 5 Blue marbles. A fair coin is flipped. If the coin comes up Heads, 2 Red and 1 White marbles are added to the urn. On the other hand, if the coin comes up Tails, 1 White marble is removed from the urn. Three random marbles are now drawn (without replacement) from the urn. (a) What is the probability that all three marbles have different colors? (b) What is the probability that all three marbles have the same color if the coin flip was Heads? (c) What is the probability that the coin flip was Heads, given that all three marbles have the same color?

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