Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities.
The mean number of oil tankers at a port city is 10 per day. The port has facilities to handle up to 14 oil tankers in a day. Find the probability that on a given day, (a) ten oil tankers will arrive, (b) at most three oil tankers will arrive, and (c) too many oil tankers will arrive.
(b) $\mathrm{P}$ (at most three oil tankers will arive $)=0.0103$
(Round to four decimal places as needed.)
(c) $P$ (too many oil tankers will arrive $)=0.0835$
(Round to four decimal places as needed.)
A. The event in part (a) is unusual
B. The event in part (b) is unusual.
C. The event in part (c) is unusual
D. None of the events are unusual.

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: \(\boxed{\text{B. The event in part (b) is unusual.}}\)