When the true incidence rate of an event in the population is "X" (say, 12%), how likely is it that the event is observed at least once in a random sample of "Y" (say, 250) subjects drawn from this population?

Re: Incidence rate in population and in random sample

If you can assume that the probability of the event is the same for each individual (unrealistic in life, but convenient and simple nonetheless), you can rely on the binomial distribution, with parameter p = 0.12, and n = 250 trials.

To simplify things, you could note that and just focus on computing , where Z is the number of subjects with the event in the sample of 250.