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.

Living Donor Transplants The mean number of organ transplants from living donors performed per day in the United States in 2020 was about 16. Find the probability that the number of organ transplants from living donors performed on any given day is (c) no more than 10. (Source: Organ Procurement and Transplantation Network)

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Step 1: Identify the type of distribution to use. Since the problem involves the mean number of events (organ transplants) occurring per day and the events are independent, 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 mean rate of occurrence is known.

Step 2: Write down the formula for the Poisson probability. The probability of observing exactly k events in a Poisson distribution is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).

Step 3: Define the problem in terms of the Poisson distribution. Here, λ = 16 (the mean number of organ transplants per day), and we are tasked with finding the probability that the number of transplants is no more than 10. This means we need to calculate P(X ≤ 10), which is the cumulative probability for k = 0, 1, 2, ..., 10.

Step 4: Use the cumulative probability formula. To find P(X ≤ 10), sum the probabilities for each value of k from 0 to 10: P(X ≤ 10) = P(X = 0) + P(X = 1) + ... + P(X = 10). For each term, substitute the values of λ and k into the Poisson formula: P(X = k) = (16^k * e^(-16)) / k!.

Step 5: Use technology or a Poisson distribution table to compute the cumulative probability. Alternatively, use statistical software or a calculator with a Poisson cumulative distribution function (CDF) to directly compute P(X ≤ 10). Once the probability is found, compare it 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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as the number of organ transplants per day. The key parameter is the mean (λ), which represents the average number of occurrences in the interval.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring, expressed as a number between 0 and 1. In the context of the Poisson distribution, this is done using the formula P(X=k) = (e^(-λ) * λ^k) / k!, where k is the number of occurrences, λ is the mean, and e is Euler's number. Understanding how to apply this formula is essential for finding the required probabilities.

Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In statistical analysis, determining whether an event is unusual helps in understanding its significance in the context of the data. For the given problem, after calculating the probability of 10 or fewer transplants, one would assess if this probability falls below the threshold to classify it as unusual.

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