No email in the next five minutes

Question

Emails arrive in your inbox at random, with an average gap of 10 minutes between them, modeled as an exponential waiting time. Right now the inbox is quiet.

What is the probability that no email arrives in the next 5 minutes?

Accepted Answer

For an exponential wait, the chance of still waiting after time t (no arrival yet) is P(T t) = e^{-\lambda t}, where \lambda is the arrival rate. "No email in the next 5 minutes" is exactly the event that the waiting time exceeds 5 minutes. The rate is one email per average gap, so \lambda = 	frac{1}{10} per minute. For t = 5 minutes, P(T 5) = e^{-\frac{1}{10}\cdot 5} = e^{-0.5} \approx 0.607. Here 5 minutes is half of one average gap, so the exponent is -0.5, giving roughly a 61% chance of a quiet five minutes. That is more than half, which fits intuition: you are asking about a stretch sho…

No email in the next five minutes

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