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 (a) exactly 12 (Source: Organ Procurement and Transplantation Network)

Verified step by step guidance

1. Step 1: Identify the type of distribution to use. Since the problem involves the mean number of events (16 transplants per day) and asks for the probability of a specific number of events (exactly 12 transplants), this is a Poisson distribution problem. The Poisson distribution is used to model the probability of a given 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: Write down the formula for the Poisson probability distribution. The probability of observing exactly k events is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the specific number of events, and e is the base of the natural logarithm (approximately 2.718).
3. Step 3: Substitute the given values into the formula. Here, λ = 16 (mean number of transplants per day) and k = 12 (specific number of transplants). The formula becomes: P(X = 12) = (16^12 * e^(-16)) / 12!.
4. Step 4: Simplify the expression. Calculate the numerator (16^12 * e^(-16)) and the denominator (12!). If using technology or a statistical calculator, input these values directly to find the probability. Alternatively, use a Poisson probability table or software to find P(X = 12) for λ = 16.
5. Step 5: Determine whether the event is unusual. An event is typically considered unusual if its probability is less than 0.05. Compare the calculated probability to this threshold to decide if observing exactly 12 transplants in a day is 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 represents 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 successes, such as the number of organ transplants performed in a day, given a specific probability.