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Problem 5 (10 points, graded on completion) (Problem on the monkey again) (1) Consider Problem 3 again but this time, instead of approximating using Poisson's paradigm, compute an exact recursive formula, considering the event E n that there is a string of k consecutive A 's in a sequence of n random letters, conditioning on the partition of events F j , j = 1 , , k and A , where F j denotes the event where the first non A -letter appears in position j , and the event A the event that the first k letters are all A 's. (2) Use Google sheets or some other computing software to calculate the probabilities for a few values of n and k using this recursive formula, and compare them to the ones obtained in the approximation of Problem 3. .

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Problem 5 (10 points, graded on completion) (Problem on the monkey again) (1) Consider Problem 3 again but this time, instead of approximating using Poisson's paradigm, compute an exact recursive formula, considering the event E n that there is a string of k consecutive A 's in a sequence of n random letters, conditioning on the partition of events F j , j = 1 , , k and A , where F j denotes the event where the first non A -letter appears in position j , and the event A the event that the first k letters are all A 's. (2) Use Google sheets or some other computing software to calculate the probabilities for a few values of n and k using this recursive formula, and compare them to the ones obtained in the approximation of Problem 3. .

- 1. Problem 5 (10 points, graded on completion) (Problem on the monkey again) (1) Consider Problem 3 again but this time, instead of approximating using Poisson's paradigm, compute an exact recursive formula, considering the event E n that there is a string of k consecutive A 's in a sequence of n random letters, conditioning on the partition of events F j , j = 1 , , k and A , where F j denotes the event where the first non A -letter appears in position j , and the event A the event that the first k letters are all A 's. (2) Use Google sheets or some other computing software to calculate the probabilities for a few values of n and k using this recursive formula, and compare them to the ones obtained in the approximation of Problem 3.