precept2

# precept2 - Computer Science 340 Reasoning About Computation...

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Computer Science 340 Reasoning About Computation Precept 2 Problem 1: Let a , b and n be natural numbers, prove that ( a + b ) n +( a - b ) n 2 is also a natural number. Problem 2: Let S = { 1 , 2 , ..., n } . Choose two random subsets A, B from all possible non-empty subsets of S . What is the probability that min( A ) = min( B ) (where min( A ) is the minimum number from the set A ). Problem 3: You are given a biased coin, that when flipped, produces Heads with unknown probability p , where 0 < p < 1. Show how a fair “coin flip” can be simulated (i.e. describe a random experiment that produces the events Heads and
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Unformatted text preview: and Tails with probability exactly 1 2 each), by looking at multiple ﬂips. Problem 4: Consider a variant of the “Monty Hall” game show discussed in class. There are k ≥ 3 doors, one of which has a grand prize behind it. You pick one of the k doors initially. The host opens one of the remaining doors to show that the prize is not there. You are now given the option of changing your initial choice. What should you do? Compute the probability of guessing the correct door for the strategy you propose....
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