Distribution for low probability events I am quite new to statistics, learning on my own, please be kind :)
I have process that should be viewed as Binomial distribution. It is PASS/FAIL type of process, basicly coin toss but with occurence of 1st outcome 99,8% and 2nd 0,2%.
Can I assume this one-sided events fall into standard Binomial distribution and use corresponding equations or is there "special" category for such one-sided processes?
Thank you.
 A: The binomial distribution with parameters n and p gives you the number of successes in n independent trials, each with a probability of success p. The probability of success, p,  can be anywhere between 0 and 1. However, when p is very large (close to 1) or very small (close to 0) and n is very large, the calculations will be extraordinarily tedious, in fact even our powerful, modern computers will struggle calculating some of them. Take for example, the probability of having 60,000 successes out of 100,000 trials: P(X=60,000), where X~Binomial(n=100,000, and p=0.998). From the Binomial PMF, you can write this probability as $P(X=60,000)= {100,000 \choose 60,000 }p^{60,000}(1-p)^{40,000} $ We know that we'll end up with a legitimate probabilty value between 0 and 1, however the journey will be a long one because ${100,000 \choose 60,000 }$ is an incredibly large number, while $p^{60,000}(1-p)^{40,000}$ is an incredibly small number.
In such cases when n is very large and p or (1-p) is very small, you'll be better served by the so-called Poisson limit or the Poisson approximation to the binomial. First, let's swap what we call a success and failure, so that p=1-0.998=0.002. Suppose X~Binomial(n,k). It can be shown that:
$$ \lim_{p\to 0, n\to \infty, np=constant} P(X=k)= \lim_{p\to 0, n\to \infty, np=constant}{n \choose k }p^{k}(1-p)^{n-k} = \frac{e^{-np(np)^k}}{k!} $$
The natural follow up question is how large would n have to be for the Poisson limit to provide a good approximation. The answer is that it depends,  but generally even with n=100, the approximation is remarkably good.
