This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Here's how to solve the problem using the Poisson distribution: The number of emails follows a Poisson distribution with an average rate of 0.2 emails per minute. The probability mass function for a Poisson distribution is given by P(X=k) = ^k e^-k!, where is the average number of events in the given interval. a) What is the probability that I get no emails in an interval of length 5 minutes? Step 1: Calculate the average rate for a 5-minute interval. The rate is 0.2 emails per minute. For a 5-minute interval, the average number of emails is: _a = 0.2 emails/minute × 5 minutes = 1 email Step 2: Calculate the probability of getting no emails (k=0) using _a = 1. P(X=0) = _a^0 e^-_a0! = 1^0 e^-10! = 1 × e^-11 = e^-1 P(X=0) ≈ 0.367879 The probability of getting no emails in a 5-minute interval is approximately 0.3679. b) What is the probability that I get more than 3 emails in an interval of length 10 minutes? Step 1: Calculate the average rate for a 10-minute interval. The rate is 0.2 emails per minute. For a 10-minute interval, the average number of emails is: _b = 0.2 emails/minute × 10 minutes = 2 emails Step 2: Calculate the probability of getting more than 3 emails. P(X > 3) = 1 - P(X 3). This means P(X > 3) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3)]. We use _b = 2 for these calculations. Step 3: Calculate P(X=0), P(X=1), P(X=2), and P(X=3). P(X=0) = 2^0 e^-20! = 1 × e^-21 = e^-2 ≈ 0.135335 P(X=1) = 2^1 e^-21! = 2 × e^-21 = 2e^-2 ≈ 0.270671 P(X=2) = 2^2 e^-22! = 4 × e^-22 = 2e^-2 ≈ 0.270671 P(X=3) = 2^3 e^-23! = 8 × e^-26 = (4)/(3)e^-2 ≈ 0.180447 Step 4: Sum the probabilities and subtract from 1. P(X 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) P(X 3) ≈ 0.135335 + 0.270671 + 0.270671 + 0.180447 = 0.857124 P(X > 3) = 1 - P(X 3) = 1 - 0.857124 = 0.142876 The probability of getting more than 3 emails in a 10-minute interval is approximately 0.1429. Send me the next one 📸