A secretary received an average of 10 emails between \( 1 \mathrm{p} . \mathrm{m} \). and \( 4 \mathrm{p} . \mathrm{m} \). on a particular day. Calculate the probability that she will receive between 11 and 16 emails at that interval of time. ( 3 marks) If the probability that she will receive at most \( m \) emails at that interval of time is 0.8645 , determine the value of \( m \).

To find the probability that the secretary receives between 11 and 16 emails in the given time interval, we can model the situation using a Poisson distribution. The average rate (\(\lambda\)) of receiving emails is 10. The Poisson probability mass function is given by: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] To find \( P(11 \leq X \leq 16) \), we can calculate: \[ P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) \] For example, to calculate \( P(X = 11) \): \[ P(X = 11) = \frac{e^{-10} \cdot 10^{11}}{11!} \] You can continue calculating each term similarly and add them together for the final probability. For the second part, where the probability that she receives at most \( m \) emails is 0.8645, we need to find \( m \) such that: \[ P(X \leq m) = 0.8645 \] Using a Poisson probability table or calculator for \( \lambda = 10 \), we find values of \( P(X \leq k) \) starting from \( k = 0 \) onwards until we reach or exceed 0.8645. By checking these cumulative probabilities, we determine that \( m \approx 13 \) is the point where the cumulative probability first exceeds 0.8645.