What you're looking at is a Poisson distribution. This assumes that the event occurs randomly but at a constant "average" rate.
For this problem it is easier to calculate the probability that the event has NOT occurred after a certain time span t. This is given by
P_not_occurring(t) = e\\^(-lambda * t)
where t would be measured in years for your problem, and lambda is a constant. Then,
P_occurring(t) = 1 - e\\^(-lambda * t).
This measures the probability of the event occurring at least once within the time span t.
You can calculate the value of lambda if you have the probability of the event occurring after one year, by substituting 1 in for t:
P_occuring(1) = 1 - e\\^(-lambda)
e\\^(-lambda) = 1 - P_occuring(1)
lambda = -ln(1 - P_occuring(1))
where ln is the natural logarithm.
In this case the probability of occurrence is always continuous (assuming that the event truly is Poisson distributed). I'm not sure what you mean by saying that it might be discrete.