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quizsol10 - CS 346 Cryptography FALL 2010 Background Quiz...

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CS 346 Cryptography FALL 2010 Background Quiz SOLUTIONS
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1. What is the numeric value of log 2 (8 50 )? log 2 (8 50 ) = 50 · log 2 8 = 50 · log 2 2 3 = 50 · 3 = 150. 2. Give a big-O estimate for each of the following functions: A. 2 n 3 + n 2 log n = O ( n 3 ) B. ( n 2 + 8)( n + 1) = O ( n 3 ) C. n ! + 2 n = O ( n !) 3. A fair die is rolled 10 times, independently. What is the probability that a 6 appears at least once? If a fair die is rolled once, the probability that a 6 does not appear is 5 6 . If a fair die is rolled 10 times, the probability that a 6 does not appear even once is ( 5 6 ) 10 . So if a fair die is rolled 10 times, the probability that a 6 appears at least once is 1 - ( 5 6 ) 10 . 4. An integer is chosen at random from the first 300 positive integers. What is the probability that the integer chosen is divisible by 2 or by 3? Let E denote the event that the integer chosen is divisible by 2, and let F denote the event that the integer chosen is divisible by 3. Then P ( E ) = 150 / 300 = 1 / 2 and P ( F ) = 100 / 300 = 1 / 3. We are looking for the probability P ( E F ). We know that P ( E F ) = P ( E ) + P ( F
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