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 (b) at least eight (Source: Organ Procurement and Transplantation Network)

Verified step by step guidance

1. Step 1: Identify the appropriate probability distribution to use. Since the problem involves the mean number of events (organ transplants) occurring per day and these events are independent, the Poisson distribution is appropriate. The Poisson distribution is defined by the parameter λ (lambda), which represents the mean number of occurrences in a fixed interval. Here, λ = 16 transplants per day.
2. Step 2: Define the probability formula for the Poisson distribution. The probability of observing exactly k events in a Poisson distribution is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events, λ is the mean, and e is the base of the natural logarithm (approximately 2.718).
3. Step 3: Translate the problem into a mathematical expression. The problem asks for the probability that the number of transplants is at least 8. This can be expressed as P(X ≥ 8). Using the complement rule, this is equivalent to 1 - P(X < 8), which can be rewritten as 1 - P(X ≤ 7).
4. Step 4: Calculate P(X ≤ 7) using the cumulative probability formula for the Poisson distribution. This involves summing the probabilities for all values of k from 0 to 7: P(X ≤ 7) = P(X = 0) + P(X = 1) + ... + P(X = 7). For each term, use the Poisson formula: P(X = k) = (λ^k * e^(-λ)) / k!.
5. Step 5: Use technology or a Poisson distribution table to compute the cumulative probability P(X ≤ 7). Subtract this value from 1 to find P(X ≥ 8). Finally, determine whether the event is unusual by comparing the probability to a threshold (e.g., 0.05). If the probability is less than the threshold, the event is considered unusual.

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. This distribution is useful for scenarios where we are interested in the number of attempts until a specific event occurs, such as finding the first successful organ transplant.

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 applicable for rare events, such as organ transplants, where the events are independent. The mean of the distribution corresponds to the average number of occurrences, making it suitable for calculating probabilities in this context.

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 defined by two parameters: the number of trials and the probability of success. This distribution is relevant when assessing the likelihood of a certain number of successful transplants out of a predetermined number of attempts.