Will the third order take over two hours?

Question

Orders arrive at a warehouse as a Poisson process at rate \lambda = 3 per hour. What is the probability that the 3rd order takes more than 2 hours to arrive?

Accepted Answer

There is a clean trick for questions about the time until the k-th arrival: turn the statement about time into a statement about a count. The 3rd order arrives after 2 hours exactly when fewer than 3 orders have arrived by the 2-hour mark. In other words, P(	ext{3rd arrival} 2) = P(N(2) \le 2), where N(2) is the number of orders in the first 2 hours. That count is Poisson with mean \mu = \lambda t = 3 	imes 2 = 6. So we add up the Poisson probabilities for 0, 1, and 2 orders: P(N \le 2) = e^{-6}\left(\frac{6^0}{0!} + \frac{6^1}{1!} + \frac{6^2}{2!}ight) = e^{-6}(1 + 6 + 18) = 25\,e^{-6} \…

Will the third order take over two hours?

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