Deriving an unknown probability distribution

Hi

I'm trying to find a probability distribution to fit the following
scenario:

I have a finite set of events that must occur in a finite interval
(continuous on the real line). For all sub-intervals of a constant
length x, I would like to know the probability that the maximum number
of events does not exceed a certain number.

I've looked at approximating this with a poisson distribution using
order statistics to get the maximum, but haven't had any success (or
the approximation going this route is just bad in general).

Does anyone have any suggestions on a solution to this?

Thanks

Tariq
.