Step 1 :The problem involves calculating probabilities using the Poisson distribution. The Poisson distribution is used to model the number of events occurring within a given time period. The mean number of oil tankers arriving at the port city per day is given as 10. This is the lambda (λ) for our Poisson distribution.
Step 2 :For part (a), we need to find the probability that exactly 10 oil tankers will arrive on a given day. The formula for the Poisson probability is: \(P(X=k) = λ^k * e^-λ / k!\) where: \(P(X=k)\) is the probability we want to find, λ is the mean number of events (in this case, the mean number of oil tankers arriving per day), k is the number of events we are interested in (in this case, 10 oil tankers), e is the base of the natural logarithm (approximately 2.71828), and k! is the factorial of k.
Step 3 :For part (b), we need to find the probability that at most 3 oil tankers will arrive on a given day. This is the sum of the probabilities that 0, 1, 2, or 3 oil tankers will arrive. We can use the same Poisson probability formula as above, but we need to calculate it for k=0, k=1, k=2, and k=3, and then sum these probabilities.
Step 4 :The calculated probabilities for parts (a) and (b) of the question are: The probability that exactly 10 oil tankers will arrive on a given day is approximately 0.1251, and the probability that at most 3 oil tankers will arrive is approximately 0.0103.
Step 5 :Final Answer: (a) The probability that exactly ten oil tankers will arrive is approximately \(\boxed{0.1251}\). (b) The probability that at most three oil tankers will arrive is approximately \(\boxed{0.0103}\).