Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.


Oil Tankers In the month of June 2021, 240 oil tankers stop at a port city. No oil tanker visits more than once. Find the probability that the number of oil tankers that stop on any given day in June is (c) more than eight.

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since the problem involves counting the number of oil tankers stopping at a port city on a given day, and the events occur over a fixed interval (days in June), the Poisson distribution is appropriate. The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space when the events occur independently and at a constant average rate.
2. Step 2: Calculate the average rate (λ) of oil tankers stopping per day. The total number of oil tankers in June is 240, and June has 30 days. Therefore, the average rate is λ = 240 / 30 = 8 oil tankers per day.
3. Step 3: Define the probability formula for the Poisson distribution. The probability of observing k events in a Poisson distribution is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).
4. Step 4: To find the probability that the number of oil tankers stopping on a given day is more than 8, calculate P(X > 8). This can be expressed as 1 - P(X ≤ 8). Use the cumulative probability formula to calculate P(X ≤ 8), which is the sum of probabilities for k = 0, 1, 2, ..., 8: P(X ≤ 8) = Σ [(λ^k * e^(-λ)) / k!] for k = 0 to 8.
5. Step 5: Use technology (e.g., a statistical calculator, software, or a Poisson distribution table) to compute P(X ≤ 8) and subtract it from 1 to find P(X > 8). Finally, compare the result to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05.

Key Concepts

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It is characterized by a constant probability of success on each trial. In the context of the question, it can be used to find the probability of a certain number of oil tankers stopping at the port on a given day, assuming each day is an independent trial.

Poisson Distribution

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for rare events. In this scenario, it can help determine the probability of more than eight oil tankers stopping at the port on any given day in June, based on the average number of tankers per day.

Binomial Distribution

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is applicable when there are two possible outcomes (success or failure) for each trial. While this distribution may not be the primary focus for the given problem, understanding it is essential for comparing it with the Poisson and geometric distributions in terms of event occurrence.