Q:

Using a Rate as a Probability

Can I use a rate for something as a probability that it will occur once?For example, I have the rate of something occurring in a given mile of driving. Can I use that rate as the probability that it occurs once in a given mile?
My thought process is if I use a Poisson distribution and set the number of occurrences to 1, then I just get the initial rate. Sorry if that didn't make sense, I don't have much of a math background. Thanks!

A:

In a Poisson process with rate $r$, the number of occurrences in an interval of length $L$ is a Poisson random variable with parameter $rL$.
The probability of exactly one occurrence in this interval is then $
rL \exp(-rL)$.  This would be $r$ if $L \exp(-rL) = 1$.   If $r \le e^{-1}$, there will be exactly two such $L$ values.  If $r > e^{-1}$, there are none.

