The exponential function and the Poisson distribution
The exponential function can be written as a series (infinite sum):
$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
The Poisson distribution is defined by the probabilities
$$p(x)=e^{-\lambda}\frac{\lambda^x}{x!}\textrm{ for } x=0,\ 1,\ 2,\ \ldots$$
Details
The exponential function can be written as a series (infinite sum):
$$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$
Knowing this we can see why the Poisson probabilities
$$p(x)=e^{-\lambda}\frac{\lambda^x}{x!}$$
add to one:
$$\sum_{x=0}^{\infty}p(x)=\sum_{x=0}^{\infty}e^{-\lambda}\frac{\lambda^x}{x!}=e^{-\lambda}\sum_{x=0}^{\infty}\frac{\lambda^x}{x!}=e^{-\lambda}e^{\lambda}=1.$$