Denote by N the outcome of rolling a six-faced fair die. Then we roll the die...

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Denote by N the outcome of rolling a six-faced fair die. Then we roll the die again N times independently. Among the last N outcomes, denote by X the number of times that the outcomes are 1. Let Y = min{X, N} (i.e. Y is the smaller one between X and N). (a) Find Pr(X 1|N = n). (b) Find Pr(X = 6). (c) Find Pr(Y6). (d) Use Bayes' theorem but not otherwise, find Pr(N = n|X = 6). (e) Are the events (X 1} and {X = 6} independent? Justify your answer. (f) Are the events {X = 6} and {N = n} independent? Justify your answer. (3 marks) (3 marks) (4 marks) (6 marks) (3 marks) (4 marks) Denote by N the outcome of rolling a six-faced fair die. Then we roll the die again N times independently. Among the last N outcomes, denote by X the number of times that the outcomes are 1. Let Y = min{X, N} (i.e. Y is the smaller one between X and N). (a) Find Pr(X 1|N = n). (b) Find Pr(X = 6). (c) Find Pr(Y6). (d) Use Bayes' theorem but not otherwise, find Pr(N = n|X = 6). (e) Are the events (X 1} and {X = 6} independent? Justify your answer. (f) Are the events {X = 6} and {N = n} independent? Justify your answer. (3 marks) (3 marks) (4 marks) (6 marks) (3 marks) (4 marks)