Already waited ten minutes for a taxi

Question

Empty taxis pass your corner as a Poisson process at rate $\lambda = 4$ per hour. You have already been waiting $10$ minutes and none has come.

Given that, what is the probability you must wait at least $15$ more minutes?

Accepted Answer

Exponential waiting times are memoryless: the process does not "remember" how long you have already waited. Whatever has happened so far, the remaining wait has the very same exponential distribution as a fresh wait starting now. Formally, P(T s + t \mid T s) = P(T t). So the 10 minutes already spent are irrelevant, and we just need the chance a fresh wait exceeds 15 minutes: P(T 	frac{15}{60}) = e^{-\lambda t} = e^{-4 	imes 1/4} = e^{-1} \approx 0.368. About a 37\% chance. Your patience so far earned you nothing statistically; a taxi is no more "due" than at the moment you arrived. Takeawa…

Already waited ten minutes for a taxi

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