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) $P$ (ten oil tankers will arrive $)=0.1251$
(Round to four decimal places as needed)
(b) $\mathrm{P}$ (at most three oil tankers will arhe) =
(Round to four decimal places as needed.)
Answer
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 \(\boxed{0.0103}\).
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 \(\boxed{0.0103}\).
