What is the variance of the number of times a 6 appears when a fair die is rolled 10 times?
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Answer:CS 2022/ MA 2201 Discrete MathematicsA term 2015Solutions for Homework 51.Exercise 28 on page 451.Solution:•First solution: We need to find the number of ways for the com-puter to select its 11 numbers, and we need to find the numberof ways for it to select its 11 numbers so as to contain the 7numbers that we choose. For the former, the number is clearlyC(80,11) =(8011). For the latter, the computer must select fourmore numbers besides the ones we choose, from the 80−7 = 73other numbers, so there areC(73,4) =(734)ways to do this.Therefore the probability that we win is the ratio(734)(8011).•Second Solution: The same answer can be obtained by countingthings in the other direction: the number of ways for us to choose 7of the computer’s predestined 11 numbers divided by the numberof ways for us to pick 7 numbers. This gives the probability(117)(807),which has the same value as before.(15 points)2. In an experiment you pick at random a bit string of length 5. Considerthe following events:E1: the bit string chosen begins with 1,E2: thebit string chosen ends with 1,E3: the bit string chosen has exactlythree 1
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