The probability of a single event over different time intervals was shown not to be linear. One can make estimates of the probabilities by counting the number of events in a given time but there is a problem with measuring small probabilities the events are not likely to be counted. One can get a better estimate of a small probability by observing over a long period of time and then use the formula to convert to a probility over a smaller period of time. It is easier to use the probabilities that the events will not occur in order to do the conversion.
The number of events do not scale in the same way. Suppose we expect on average n=RΔt events in time Δt. The same rate applies to each Δt so for ΔT=mΔt the number expected is N=mn=mRΔt=RΔT. The rate, R, is independent of the size of the period of time involved.
The number of earthquakes of different magnitude which occur in a given period of time can be treated as an
urn problem.
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