During a professor's office hours,students arrive,on average,every ten minutes.Assume that the distribution of the time between arrivals follows an exponential distribution.Suppose that a student has just left. What is the probability that the professor has more than 20 minutes before the next student shows up?
A)0.8187
B)0.8647
C)0.1353
D)0.1813

Question

Quizplus · Accepted Answer

Let X be the time between arrivals of two consecutive students. Then X follows an exponential distribution with parameter λ = 1/10 (since students arrive on average every 10 minutes). The probability that the professor has more than 20 minutes before the next student shows up is equivalent to the probability that X > 20, given that a student has just left. Using the memoryless property of the exponential distribution, we have:

P(X > 20 | X > 0) = P(X > 20)/P(X > 0) = (e^(-λx=20))/e^(-λx=0) = e^(-2) ≈ 0.1353

Therefore, the best choice is (C).

During a Professor's Office Hours,students Arrive,on Average,every Ten Minutes