Waiting on the next Geiger click

Question

A radioactive sample produces Geiger clicks as a Poisson process at rate $\lambda = 2$ per minute. It has now been $3$ minutes since the last click.

Given that, what is the probability the next click is more than $90$ seconds away?

Accepted Answer

Radioactive decay is the textbook example of memorylessness: the atom does not age, so the time to the next click has no memory of how long you have already waited. The 3 quiet minutes are irrelevant; the remaining wait is a fresh exponential. Ninety seconds is 1.5 minutes, so P(T 1.5) = e^{-\lambda t} = e^{-2 	imes 1.5} = e^{-3} \approx 0.0498. About a 5\% chance. A long silent stretch does not make a click "due"; nature simply resets. Takeaway. Memorylessness means a long quiet spell tells you nothing about when the next event lands; the remaining wait always starts over as a fresh exponen…

Waiting on the next Geiger click

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