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Say I have an average amount of time X seconds between instances of an event happening with a total time frame Y seconds. What is the probability of the event happening at any given moment in Y?

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Here is a simple model that you can use if you are measuring the time of occurrence of each event in integer number of seconds (and not fractions of seconds), and at most one event can occur each second. The probability of occurrence of an event at any given second is $p = 1/X$, the probability of non-occurrence is $q = 1-p = 1 - 1/X = (X-1)/X$. The waiting time for the next occurrence is called a geometric random variable with parameter $p$, and the average waiting time is $1/p = X$ seconds as you have measured it.

There are a lot of assumptions that need to be made to fully justify this model but I won't go into them for now.

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