Waiting past eight minutes for a chat

Question

On a support desk, the time until the next customer message arrives is exponentially distributed with a mean gap of 4 minutes. Exponential timing is the natural model for purely random arrivals with no schedule.

What is the probability that you wait more than 8 minutes for the next message?

Accepted Answer

The exponential distribution has one handy fact: the chance of still waiting after time t (nothing has happened yet) is P(T t) = e^{-\lambda t}, where \lambda is the rate, the average number of events per unit time. In words, waiting-longer chances shrink smoothly and exponentially as time passes. The rate is one event per average gap. With a mean gap of 4 minutes, \lambda = 	frac{1}{4} per minute. For t = 8 minutes, P(T 8) = e^{-\frac{1}{4}\cdot 8} = e^{-2} \approx 0.135. A neat mental picture: 8 minutes is exactly two average gaps, so the exponent is -2, giving about a 13.5% chance of stil…

Waiting past eight minutes for a chat

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