4. A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is...

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4. A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. a. Suppose the monkey types n letters. Let X₁ be an indicator random variable and X₁=1 if "proof" appears at index i (assuming i<=n-5). Otherwise, X₁-0. What is the probability that Pr{X₁=1}? b. If the monkey types 1,000,000 letters, what is the expected number of times the sequence "proof" appears? 4. A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. a. Suppose the monkey types n letters. Let X₁ be an indicator random variable and X₁=1 if "proof" appears at index i (assuming i<=n-5). Otherwise, X₁-0. What is the probability that Pr{X₁=1}? b. If the monkey types 1,000,000 letters, what is the expected number of times the sequence "proof" appears? 4. A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. a. Suppose the monkey types n letters. Let X₁ be an indicator random variable and X₁=1 if "proof" appears at index i (assuming i<=n-5). Otherwise, X₁-0. What is the probability that Pr{X₁=1}? b. If the monkey types 1,000,000 letters, what is the expected number of times the sequence "proof" appears? 4. A monkey types on a 26-letter keyboard that has lowercase letters only. Each letter is chosen independently and uniformly at random from the alphabet. a. Suppose the monkey types n letters. Let X₁ be an indicator random variable and X₁=1 if "proof" appears at index i (assuming i<=n-5). Otherwise, X₁-0. What is the probability that Pr{X₁=1}? b. If the monkey types 1,000,000 letters, what is the expected number of times the sequence "proof" appears?

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