Poisson and Approximating Binomial

[AaronBrown]

In addition to the excellent replies by Siegmund and phzon, I'd add that the Poisson is most useful not only when the events are rare, but when the sample size is small enough that you only expect a few successes. In your case, with 300 expected successes, you'll get similar answers with the Normal and Poisson approximations.

Another case in which the Poisson is useful is when you don't know the number of trials. For example, suppose in 2005, 10 people have died of heart attacks immediately after getting holes in one in golf. I'm willing to assume these are independent events. The long term average number is 5 per year. But I don't know how many people get holes in one in golf. Nevertheless, because fatal heart attacks are reasonably rare over short intervals of time, I can assume that the number per year is a Poisson distribution with mean and variance of 5. There is only a 3% chance of observing 10 or more occurrances in that case. So, this is significant evidence that something has changed (more people playing golf, more holes in one, sicker people playing golf, more heart attacks; something).

The interesting thing is I don't need any more information than the long run average, and that the events are independent and rare.