pset10_soln

pset10_soln - 6.042/18.062J Mathematics for Computer...

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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science April 26, 2005 Srini Devadas and Eric Lehman Problem Set 10 Solutions Due: Monday, May 2 at 9 PM Problem 1. Justify your answers to the following questions about independence. (a) Suppose that you roll a fair die that has six sides, numbered 1 , 2 , . . . , 6 . Is the event that the number on top is a multiple of 2 independent of the event that the number on top is a multiple of 3? Solution. Let A be the event that the number on top is a multiple of 2, and let B be the event that the number on top is a multiple of 3. We have: 3 2 1 Pr ( A ) Pr ( B ) = = = Pr ( A B ) 6 6 6 Therefore, these events are independent. (b) Now suppose that you roll a fair die that has four sides, numbered 1 , 2 , 3 , 4 . Is the event that the number on top is a multiple of 2 independent of the event that the number on top is a multiple of 3? Solution. As before, let A be the event that the number on top is a multiple of 2, and let B be the event that the number on top is a multiple of 3. Now, however, we have: 2 1 1 Pr ( A ) Pr ( B ) = = 4 4 8 But: Pr ( A B ) = 0 Since these results disagree, the events are not independent. (c) Now suppose that you roll a fair die that has eight sides, numbered 1 , 2 , . . . , 8 . Again, is the event that the number on top is a multiple of 2 independent of the event that the number on top is a multiple of 3? Solution. As before, let A be the event that the number on top is a multiple of 2, and let B be the event that the number on top is a multiple of 3. This time, we have: 4 2 1 Pr ( A ) Pr ( B ) = = 8 8 8 And: Pr ( A B ) = 1 / 8 Therefore, these events are independent. 2 Problem Set 10 (d) Finally, suppose that you roll the fair, eight-sided die again. Let the random variable X be the remainder when the number on top is divided by 2, and let the random variable Y be the remainder when the number on top is divided by 3. Are the random variables X and Y independent? Solution. First, lets tabulate the values of X and Y : die roll X Y 1 1 1 2 0 2 3 1 0 4 0 1 5 1 2 6 0 0 7 1 1 8 0 2 Working from the table, we have: 2 Pr ( X = 1 Y = 1) = 8 But: 4 3 Pr ( X = 1) Pr ( Y = 1) = 8 8 3 = 16 Since these results conict, the random variables are not independent. Problem 2. Philo T. Megabrain, a noted parapsychology researcher, has discovered an amazing phenomenon! He puts a psychic on each side of an opaque, soundproof barrier. Each psychic rolls a fair die, looks at it, and tries to guess what number came up on the other die by telepathy. Since the dice are fair and independent, the psychics should guess correctly only 1 time in 6. However, after extensive testing, Philo has discovered that they actually do slightly better....
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This note was uploaded on 02/08/2011 for the course EECS 6.042 taught by Professor Dr.ericlehman during the Spring '11 term at MIT.

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pset10_soln - 6.042/18.062J Mathematics for Computer...

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