# How to choose between poisson and binomial distributions

I don't get this thing... I know that binomial distribution is used to know the probability of a X v.a. that sounds like this: X = "the probability of having 4 blue balls doing 10 extraction from a chest containing 7 blue and 40 white", and I know that poisson distribution and binomial distribution are really similar for lim_(p*n)->0(F(x)) (when p*n are really small...). I'm reading everywhere that the distribution of poisson is used a lot to approximate a big binomial... but what's its real purpose? what's the cases in wich I must use poisson and not binomial? (beyond the approximation case).

• The Poisson distribution can be derived from the binomial distribution. The Poisson is nothing more than the limiting case of the Binomial where n is large and p is small. – Islands Nov 15 '13 at 21:41
• I would like to point out that a distribution that can be well approximated by another one is not immediately useless. – M Turgeon Nov 15 '13 at 23:10
• @MTurgeon but is Poisson distribution needless for small n*p values? – user2993157 Nov 15 '13 at 23:28
• Short answer: no. Why would a distribution be useless? The Poisson and the binomial distributions model different situations (even if there are similarities). Why would one of them be useless? – M Turgeon Nov 17 '13 at 18:17
• @MTurgeon I mean, there's a case in which Poisson is needed but not as an approximation of the Binomial? – user2993157 Nov 18 '13 at 10:06