A detector click within one second

Question

A particle detector produces clicks at random times, exponentially distributed with an average gap of 5 seconds. A click just occurred.

What is the probability that the next click comes within 1 second?

Accepted Answer

This asks for the chance the wait is short (at most 1 second), which is the complement of waiting longer. Start from the survival rule, the chance of still waiting after time t: P(T t) = e^{-\lambda t}. The chance the event happens by time t is whatever is left over: P(T \le t) = 1 - e^{-\lambda t}. The rate is one click per average gap, so \lambda = 	frac{1}{5} per second. For t = 1 second, P(T \le 1) = 1 - e^{-\frac{1}{5}\cdot 1} = 1 - e^{-0.2} \approx 1 - 0.819 = 0.181. One second is only a fifth of the average gap, so it is unsurprising that a click arrives that quickly only about 18% of…

A detector click within one second

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