Question

A post office has 2 clerks. Alice enters the post office while 2 other customers, Bob and Claire, are being served by the 2 clerks. She is next in line. Assume that the time a clerk spends serving a customer has the Expo (lambda) distribution. a) What is the probability that Alice is the last of the 3 customers to be done being served? b) What is the expected total time that Alice needs to spend at the post office?

Answer 1

To find the probability that Alice is the last of the 3 customers to be done being served, subtract the probability that both Bob and Claire finish before Alice from 1. The expected total time that Alice needs to spend at the post office is the sum of the expected times for Bob and Claire to finish.

Step-by-step explanation:

To answer this question, we need to understand the properties of the Exponential distribution and how it relates to arrival and service times. The Exponential distribution is often used to model the time between events. In this case, the time a clerk spends serving a customer follows an Exponential distribution with an average of four minutes.

a) Probability that Alice is the last of the 3 customers

Let's consider the probabilities of the other two customers finishing before Alice. The probability that Bob finishes before Alice is the probability that a randomly selected customer takes less than four minutes to be served, which is equal to 1 - e^(-λx). Similarly, the probability that Claire finishes before Alice is 1 - e^(-λx). However, since Bob and Claire are being served by different clerks, their serving times are independent, so the probability that both finish before Alice is (1 - e^(-λx))^2. Therefore, the probability that Alice is the last of the three customers to be done being served is 1 - (1 - e^(-λx))^2.

b) Expected total time that Alice needs to spend at the post office

The expected total time that Alice needs to spend at the post office is the sum of the time the other two customers take to finish before Alice. The expected time for Bob to finish is 1/(2λ) and the expected time for Claire to finish is 1/(2λ). Therefore, the expected total time that Alice needs to spend at the post office is (1/(2λ)) + (1/(2λ)).

Answer 2

Alice has a 0.5 probability of being the last of the three customers to be served, and the expected total time she spends at the post office is 8 minutes.

Step-by-step explanation:

The question is asking for two pieces of information regarding an exponential distribution of the time clerks at a post office spend with a customer. We are given that Alice, Bob, and Claire are in the post office with Alice being the next in line.

a) Since the service times are exponentially distributed, the memoryless property implies that the probability of any one clerk finishing first is independent of how long they've already been serving the current customer. This means Alice has an equal chance of being served by either of the clerks, so the probability that she will be the last to finish being served is 0.5, assuming that service times are independent and identical.

b) The expected total time for Alice is the sum of her expected waiting time and expected service time. Since she is next in line, her expected waiting time is the average service time of a clerk, which is equal to the parameter λ. As stated in Example 5.7, the average time is four minutes. So, her expected wait time is 4 minutes plus the expected service time, which is also 4 minutes, making the total expected time 8 minutes.