Let $X$ be distributed as $Bin(n,p)$, a binomial random variable with parameters $n$ and $p$. Suppose

where $\lambda$ is a positive real constant, then $X$ is asymptotically distributed as $Poisson(\lambda)$, a Poisson distribution with parameter $\lambda$.

Basically, when the size of the population $n$ is very large and the occurrence of certain event $A$ is rare, where $p$, the probability of $A$ is very small, the binomial random variable $X$ can be approximated by a Poisson random variable.

Sketch of Proof. Let $X\sim Bin(n,p)$. So

As $n\rightarrow\infty$,

and

Therefore,

Example. Suppose in a given year, the number of fatal automobile accidents has a binomial distribution for a particular insuarance company with five hundred automobile insurance policies. On average, there is one policy out of the five hundred that will be involved in a fatal crash. What is the probability that there will be no fatal accidents (out of five hundred policies) in any particular year?

Solution. If $X$ be the number of fatal accidents in a year from a population of 500 auto policies, then $X\sim Bin(n,p)$ with $n=500$ and $p=1/500$. $\lambda=500\times 1/500=1$ and so

Using the binomial distribution, we have