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No Rush! Your help is truly appreciated! 4. (20 points) You choose at random a binary string of length 8. Consider the following events: e

No Rush! Your help is truly appreciated!

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4. (20 points) You choose at random a binary string of length 8. Consider the following events: e E;: the binary string chosen ends with 1. e E,: the binary string chosen begins with 00. Ej3: the binary string chosen has exactly five 1's. (a) (4 points) Find p(E3|E,). Solution: (4 points) Find p(E;|E3). g Solution: | 0O - (4 points) Find p(E4|E, N E3). Solution: (d) (4 points) Determine whether E, and E; are independent. Solution: (e) (4 points) Determine whether E; and E; are independent. Solution: 5. (20 points) Suppose Ryan has 10 red and 6 blue ducks. Assume all ducks are distinguishable from each other. Keep answers in Permutation/Combination notation. (a) (10 points) Ryan wants to arrange the ducks so that no blue ducks are next to each other. If he has chosen a random arrangement of the ducks, what is the probability that none of the blue ducks are next to each other? Solution: (b) (10 points) Now imagine from the 16 ducks there are 4 ducks with special golden bottoms, two of them are blue ducks, and the other two are red ducks. If Ryan randomly picks 6 ducks from a bag containing all the ducks, what is the probability he selects exactly 1 blue duck with a golden bottom? Solution

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