Two quiet years for earthquakes

Question

Major earthquakes strike a fault region as a Poisson process at rate \lambda = 1.5 per year. What is the probability of no major earthquake for the next 2 years?

Accepted Answer

"No earthquake for the next 2 years" is the same as "the wait for the next earthquake is longer than 2 years." That wait is exponential, so P(T t) = e^{-\lambda t}. With \lambda = 1.5 per year and t = 2 years, P(T 2) = e^{-1.5 	imes 2} = e^{-3} \approx 0.0498. So only about a 5\% chance of a two-year calm. Equivalently, the number of quakes in two years has mean \mu = \lambda t = 3, and the chance of zero is e^{-3}, the same answer. Takeaway. "Nothing happens for time t" equals "the next event waits longer than t"; both equal e^{-\lambda t}.

Two quiet years for earthquakes

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