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.
(a) $\mathrm{P}$ (ten oil tankers will arrive $)=0.1251$
(Round to four decimal places as needed.)
(b) $P$ (at most three oil tankers will arivc) $=0.0103$
(Round to four decimal places as needed.)
(c) $\mathrm{P}$ (too many oil tankers will arrive) =
(Round to four decimal places as needed.)

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 \(\boxed{0.0835}\).