Twenty minutes since the last call

Question

Calls reach a small office as a Poisson process at rate $\lambda = 5$ per hour. It has been $20$ minutes since the last call.

Given that quiet stretch, what is the probability the next call arrives within the next $6$ minutes?

Accepted Answer

The wait between Poisson calls is exponential and therefore memoryless: the 20 silent minutes do not change the distribution of the remaining wait. We just ask whether a fresh wait is shorter than 6 minutes. For "within t," use one minus the exponential tail. Six minutes is 	frac{6}{60} = 	frac{1}{10} of an hour, so \lambda t = 5 	imes 	frac{1}{10} = 0.5 and P(T \le 	frac{1}{10}) = 1 - e^{-0.5} \approx 1 - 0.6065 = 0.393. About a 39\% chance. The dry spell does not make a call any more likely to come "soon" than usual. Takeaway. Because the wait is memoryless, a recent quiet period leave…

Twenty minutes since the last call

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