Step 1 :The problem is asking for the probability of certain events occurring given a Poisson distribution with a mean of 10. The Poisson distribution is often used to model the number of times an event occurs in a fixed interval of time or space. In this case, the event is the arrival of oil tankers at a port city, and the interval is one day.
Step 2 :The first part of the question asks for the probability that exactly 10 oil tankers will arrive in a day. This can be calculated using the formula for the Poisson distribution: \(P(X=k) = (λ^k * e^-λ) / k!\) where λ is the mean number of events (in this case, 10), k is the number of events we are interested in (in this case, 10), and e is the base of the natural logarithm (approximately 2.71828).
Step 3 :The second part of the question asks for the probability that at most 3 oil tankers will arrive in a day. This is the sum of the probabilities that 0, 1, 2, or 3 oil tankers will arrive. Each of these probabilities can be calculated using the formula for the Poisson distribution.
Step 4 :The third part of the question asks for the probability that more than 14 oil tankers will arrive in a day. This is 1 minus the sum of the probabilities that 0 through 14 oil tankers will arrive. Each of these probabilities can be calculated using the formula for the Poisson distribution.
Step 5 :Finally, the question asks whether these events are unusual. An event is typically considered unusual if its probability is less than 0.05.
Step 6 :Calculating the probabilities, we get \(P(A) = 0.1251\), \(P(B) = 0.0103\), and \(P(C) = 0.0835\).
Step 7 :Comparing these probabilities to the threshold of 0.05, we find that event A is not unusual (since \(P(A) > 0.05\)), event B is unusual (since \(P(B) < 0.05\)), and event C is not unusual (since \(P(C) > 0.05\)).
Step 8 :Final Answer: \(\boxed{\text{B. The event in part (b) is unusual.}}\)