Unformatted text preview: very useful fact known as the union bound: Pr( E 1 ∪ E 2 . . . E n ) ≤ Pr( E 1 ) + Pr( E 2 ) + . . . Pr( E n ) , where E 1 , E 2 , . . . E n are events associated with the same sample space. Prove it by induction and use the base case n = 2. Problem 7.7, 5 points. Here is a sample proof that contains an error. Explain why the proof is not correct. Theorem: If a  b and b  c , then a  c . Proof: Since a  b , b = ak . Since b  c, c = bk . Therefore c = bk = ( ak ) k = ak 2 . Therefore a  c . Problem 7.8, 5 points. Prove that if n is an integer that is not a multiple of 3, then n 2 ≡ 1 mod 3. Problem 7.9, 5 points. Suppose that all you know about high tide is that its expected height is 1 meter, and the height is always nonnegative. What can we say about the probability that you see high tide more than 2 meters?...
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 Spring '05
 HUANG
 Probability theory, Euclidean algorithm, Mathematical proof

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