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 interval. The formula for the Poisson probability is: \(P(x; μ) = (e^{-μ}) * (μ^{x}) / x!\) where: \(P(x; μ)\) is the Poisson probability, \(e\) is the base of the natural logarithm, \(μ\) is the mean number of successes that result from the experiment, \(x\) is the actual number of successes that result from the experiment, and \(x!\) is the factorial of x.
Step 2 :For part (a), we need to find the probability that exactly 10 oil tankers will arrive. We can use the Poisson distribution formula with \(μ = 10\) (the mean number of oil tankers per day) and \(x = 10\) (the number of oil tankers we want to find the probability for). The probability that ten oil tankers will arrive is \(0.1251\).
Step 3 :For part (b), we need to find the probability that at most 3 oil tankers will arrive. This is the sum of the probabilities that 0, 1, 2, or 3 oil tankers will arrive. We can use the Poisson distribution formula with \(μ = 10\) and \(x = 0, 1, 2, 3\), then sum these probabilities. The probability that at most three oil tankers will arrive is \(0.0103\).
Step 4 :For part (c), we need to find the probability that more than 14 oil tankers will arrive. This is 1 minus the sum of the probabilities that 0 through 14 oil tankers will arrive. We can use the Poisson distribution formula with \(μ = 10\) and \(x = 0\) through 14, sum these probabilities, and subtract from 1. The probability that too many oil tankers will arrive is \(0.0835\).
Step 5 :Final Answer: (a) The probability that ten oil tankers will arrive is \(\boxed{0.1251}\). (b) The probability that at most three oil tankers will arrive is \(\boxed{0.0103}\). (c) The probability that too many oil tankers will arrive is \(\boxed{0.0835}\).