A service engineer receives on average seven calls in a 24-hour period. Calculate the probability that in a 24 -hour period the engineer receives (a) seven calls (b) eight calls (c) six calls (d) fewer than three calls

Question: Calculate the probabilities of a service engineer receiving 7 calls, 8 calls, 6 calls, and fewer than 3 calls in a 24-hour period, given that the engineer receives, on average, 7 calls in that time frame. Answer: To calculate these probabilities, use the Poisson Distribution formula as follows: 1. Probability of receiving 7 calls: P(X = 7) = (e^(-7) * 7^7) / 7! Evaluate this expression to obtain the probability of receiving seven calls. 2. Probability of receiving 8 calls: P(X = 8) = (e^(-7) * 7^8) / 8! Evaluate this expression to obtain the probability of receiving eight calls. 3. Probability of receiving 6 calls: P(X = 6) = (e^(-7) * 7^6) / 6! Evaluate this expression to obtain the probability of receiving six calls. 4. Probability of receiving fewer than 3 calls: P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) Calculate the probabilities for k=0, 1, and 2 and add them together: P(X = 0) = (e^(-7) * 7^0) / 0! P(X = 1) = (e^(-7) * 7^1) / 1! P(X = 2) = (e^(-7) * 7^2) / 2! Evaluate these expressions and add them together to obtain the probability of receiving fewer than three calls.