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cps102-hw3 - CPS 102 Discrete Mathematics Assignment 3 Due...

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CPS 102: Discrete Mathematics Instructor: Bruce Maggs Assignment 3 Due: Monday, October 24 th , 2011 Note: You cannot discuss/collaborate with any other person other than the instructor or TA. And, when solving for any question below please show your work! 1 Counting (5 points) A college has 10 basketball players. A team of 5 players, of which 1 is a captain, will be selected out of these 10 players. How many ways are there to form the team? 2 Probability (5 points) What is the probability that a five-card poker hand contains the ace of hearts? 3 Counting (5 points) All PhD students in the CS department at Duke have to receive a Quals Pass in four of the seven des- ignated courses spread across three streams – AI / Num. Analysis : { CPS 250 , CPS 270 , CPS 271 } , Theory : { CPS 230 , CPS 240 } and Systems : { CPS 210 , CPS 220 } . However, they cannot skip a stream completely and they can only take one of { CPS 270 , CPS 271 } . In how many ways can the PhD students choose four courses of the designated seven? 4 Probability (5 points) What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads? 5 Counting (5 points) How many bit strings of length 8 contain either four consecutive 0s or four consecutive 1s? 6 Probability (5 points) Provide an example that shows that the variance of the sum of two random variables is not neces- sarily equal to the sum of their variances when the random variables are not independent.
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