Problems in Probability

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Fall 2011 Strategic Practice 6: Section 1 (Exponential Distribution and Memorylessness) - Question 4
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 Exponential() distribution.
(a) What is the probability that Alice is the last of the 3 customers to be done being served. Hint: no integrals are needed.
(b) What is the expected total time that Alice needs to spend at the post office?
Solution: (a) Alice begins to be served when either Bob or Claire leaves. By the memoryless property, the additional time needed to serve whichever of Bob or Claire is still there is Exponential(lambda). The time it takes to serve Alice is also Exponential(lambda), so by symmetry the probability is 1/2 that Alice is the last to be done being served. (b) 1/(2 x lambda) waiting time + 1/lambda serving time = 3/(2 x lambda).
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