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 (a) exactly eight

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since we are dealing with the number of oil tankers stopping at a port city per day, and the total number of tankers in the month is given, this is a Poisson distribution problem. 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.
3. Step 3: Write the formula for the Poisson probability mass function (PMF). The formula is: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (in this case, the number of oil tankers stopping on a given day), λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).
4. Step 4: Substitute the values into the formula. Here, k = 8 (since we are finding the probability of exactly 8 tankers stopping), and λ is the average rate calculated in Step 2. Plug these values into the formula: P(X = 8) = (λ^8 * e^(-λ)) / 8!.
5. Step 5: Determine whether the event is unusual. An event is typically considered unusual if its probability is less than 0.05. After calculating the probability in Step 4, compare the result to 0.05 to determine if it is unusual. If convenient, use a statistical calculator or software to compute the exact probability.

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 specific number of successes (e.g., oil tankers stopping) over a fixed number of trials (days).

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 calculate the probability of a certain number of oil tankers stopping at the port on a given day, based on the average daily rate derived from the total number of tankers over the month.

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. In this case, it could be used if we were interested in the number of days with a specific number of tankers stopping, rather than the total number of tankers in a day.