Question

Two individuals, \(A\) and \(B\), both require kidney transplants. If she does not receive a new kidney, then \(A\) will die after an exponential time with rate \(\mu_{A}\), and \(B\) after an exponential time with rate \(\mu_{B} .\) New kidneys arrive in accordance with a Poisson process having rate \(\lambda\). It has been decided that the first kidney will go to \(A\) (or to \(B\) if \(B\) is alive and \(A\) is not at that time) and the next one to \(B\) (if still living). (a) What is the probability that \(A\) obtains a new kidney? (b) What is the probability that \(B\) obtains a new kidney?

Short answer

The probabilities that A and B get kidneys are: \[P_A = 1 - e^{-\lambda \mu_A}\] \[P_B = (1 - e^{-\lambda \mu_B}) \times (1 - e^{-\lambda \mu_A})\]

Step-by-step solution

Understand the problem and define variables

In the given problem, we have two individuals A and B who need kidneys. Kidneys arrive as a Poisson process with rate \(\lambda\). A will die after an exponential time with rate \(\mu_A\), and B will die after an exponential time with rate \(\mu_B\). We need to find the probability that A and B obtain a kidney. Let's denote the probability that A obtains a kidney as \(P_A\), and the probability that B obtains a kidney as \(P_B\).

Probability that A gets a kidney

The kidney arrives as a Poisson process with rate \(\lambda\). The probability that A gets a kidney is the probability that at least one kidney arrives before A dies. The time until A dies is an exponentially distributed random variable with rate \(\mu_A\). Since the two processes are independent, we can use the memoryless property of the exponential distribution. The probability that a kidney arrives before A dies is the probability that a kidney arrives within the time interval of A's death. Mathematically, the probability that A gets a kidney is given by: \[P_A = P(\text{kidney arrives before }\mu_A \text{ time units}) = 1 - P(\text{kidney arrives after }\mu_A \text{ time units})\] Here, we have used the complement rule to simplify the expression.

Calculate the probability that A gets a kidney

We need to find the probability of a kidney arriving after \(\mu_A\) time units. Since the arrival of kidneys follows a Poisson process with rate \(\lambda\), we can use the survival function of the Poisson process: \[P(\text{kidney arrives after }\mu_A \text{ time units}) = e^{-\lambda \mu_A}\] Now, we can substitute this expression in the equation from step 2 to find the probability that A gets a kidney: \[P_A = 1 - e^{-\lambda \mu_A}\]

Probability that B gets a kidney

To calculate the probability that B gets a kidney, we first need to consider the fact that A must receive a kidney before B. After A gets a kidney, the probability that a kidney arrives before B dies follows the same logic as we used for A. Given that A got a kidney, the probability that a kidney arrives before \(\mu_B\) time units is: \[P(\text{kidney arrives before }\mu_B \text{ time units} \vert \text{ A got kidney}) = 1 - P(\text{kidney arrives after }\mu_B \text{ time units } \vert \text{ A got kidney})\] and the conditional probability that a kidney arrives after \(\mu_B\) time units given that A got a kidney is: \[P(\text{kidney arrives after }\mu_B \text{ time units} \vert \text{ A got kidney}) = e^{-\lambda \mu_B}\]

Calculate the probability that B gets a kidney

Now, we can use the conditional probability to find the probability that B gets a kidney: \[P_B = P(\text{kidney arrives before }\mu_B \text{ time units} \vert \text{ A got kidney}) \times P_A = (1 - e^{-\lambda \mu_B}) \times (1 - e^{-\lambda \mu_A})\] So, the probability that B gets a kidney is: \[P_B = (1 - e^{-\lambda \mu_B}) \times (1 - e^{-\lambda \mu_A})\] The probabilities that A and B get kidneys are: \[P_A = 1 - e^{-\lambda \mu_A}\] \[P_B = (1 - e^{-\lambda \mu_B}) \times (1 - e^{-\lambda \mu_A})\]

Key concepts

Poisson Process

The Poisson process is a mathematical concept used to model events that occur randomly over a given period of time. It's a very useful tool for understanding phenomena where events happen at a certain rate but independently from when the previous events happened. This type of process applies perfectly to situations like receiving donations of organs for transplant.

When a textbook problem talks about kidneys arriving according to a Poisson process, it means that we can make predictions about the probability of a kidney becoming available within a certain time frame, given a rate \(\lambda\). For instance, if kidneys are donated at a rate of 2 per month, you might use the Poisson process to calculate the likelihood of 3 kidneys arriving in the next two weeks.

Mathematically, the probability of exactly \( k \) events occurring in an interval can be given by the Poisson probability mass function. This function is used extensively in the medical field, for queueing theory, telecommunications, and other areas where random events are timed.

Exponential Distribution

The exponential distribution is closely tied to the Poisson process and is often used in the analysis of how much time will pass before an event occurs. It is an appropriate model for a waiting time between Poisson distributed events, and is memoryless, meaning the probability of an event occurring in the future is the same regardless of the amount of time that has already passed.

For example, if the life expectancy of a patient waiting for a transplant is represented by an exponential distribution with rate \(\mu_A\) for patient \(A\), it indicates the amount of time we expect to wait until patient \(A\) receives a kidney. It is continuous and defined for \(x > 0\), with its probability decreasing exponentially as \(x\) increases. This creates a steep decay in the probability of \(A\) surviving long enough to get a transplant as time passes, capturing the urgency of transplant timing.

In our context, we use the exponential distribution to determine the likelihood of a patient surviving until a kidney arrives, which aids in understanding and calculating associated probabilities. The function's simplicity and the memoryless property make it a powerful tool in scenario analysis and predictive modeling in healthcare settings.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is crucial when dealing with multiple events where the outcome of one affects the other. It allows us to update probabilities as we obtain new information.

In the context of our kidney transplant problem, to find the probability that \(B\) gets a kidney, we first need to know if \(A\) received one, because the order of allocation affects the outcome for \(B\). If \(A\) has already received a kidney, we use conditional probability to reassess the chances for \(B\).

The formula for conditional probability is \(P(B|A) = \frac{P(A \cap B)}{P(A)}\), where \(B|A\) represents the event \(B\) given that event \(A\) has happened. In our exercise, this translates to refining the probability that \(B\) gets a kidney based on the event that \(A\) has already received one, allowing us to evaluate the dependent outcomes progressively. Understanding conditional probability is essential for accurate risk assessment, particularly in sequential decisions or allocations, such as the distribution of limited medical resources like organs for transplant.