Algorithm to generate Poisson and binomial random numbers?

i've been looking around, but i'm not sure how to do it.

i've found this page which, in the last paragraph, says:

A simple generator for random numbers taken from a Poisson distribution is obtained using this simple recipe: if x₁, x₂, ... is a sequence of random numbers with uniform distribution between zero and one, k is the first integer for which the product x₁ · x₂ · ... · x_k+1 < e^-λ

i've found another page describing how to generate binomial numbers, but i think it is using an approximation of poisson generation, which doesn't help me.

For example, consider binomial random numbers. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads on any single toss. If you generate N uniform random numbers on the interval (0,1) and count the number less than p, then the count is a binomial random number with parameters N and p.

i know there are libraries to do it, but i can't use them, only the standard uniform generators provided by the language (java, in this case).