Why use Poisson distribution

Question

I learned from my textbook that Poisson Distribution was invented to approximate binomial distribution, so two questions arose:

1. How to get Poisson distribution from binomial distribution ? It doesn't make sense for me to do so although I know how to get binomial distribution from Poisson distribution using some approximation technique.
2. Since we still can't calculate $k!$ precisely in a reasonable amount of time, what's the point of choosing Poisson distribution over binomial distribution ?

Answer 1

The Poisson distribution can be seen as a limit case of the binomial for $n$ large ($n \to \infty$) and $p$ a small fraction of $n$, so e.g. if we have $n=1000$ and $p=\frac{1}{1000}$ we can approximate $X$ with Poisson with rate $1$. Wikipedia has more explanation. I won't derive the limit of the binomial here, but it's related to $(1+\frac{c}{n})^n \to e^c$ as $n \to \infty$...