Why does this probability converge towards $1-\frac{1}{e}$?

Question

Consider this simple problem:

If an event has a $1/N$ chance of success, what's the probability of having at least one success after $N$ events?

Basically, what's the chance of getting a $20$ if you roll a D20 $20$ times?

That probability $P(N)$ can be expressed as: $$P(N) = 1 - \left(\frac{N-1}{N}\right)^{N}$$

I found that this seems to converge towards $1-\frac{1}{e}$. Why is that?

I'm having trouble understanding what this means. Why does it converge at all? If something has a one in a million chance of happening and you try a million times, shouldn't your chance to succeed be higher that $63\%$? And why this value specifically?

Answer 1

We have

$$P(N) = 1 - \left(\frac{N-1}{N}\right)^{N}=1 - \left(1-\frac{1}{N}\right)^{N}=1 -\frac1{ \left(1-\frac{1}{N}\right)^{-N}}\to 1-\frac1e$$

since

$$\lim_{x\to \pm \infty} \left(1+\frac1x\right)^x \to e$$

Refer to the related

• Proving $\lim \limits_{n\to +\infty } \left(1+\frac{x}{n}\right)^n=\text{e}^x$.

Answer 2

This is more generally related to the fact that the Binomial distribution on $N$ events with success probability $\lambda/N$ converges in distribution to a Poisson distribution with rate $\lambda$. In your case, $\lambda=1$, and for Poisson with rate $\lambda$, $1-P_\lambda(X=0)=1-\frac{1}{e^\lambda}=1-1/e$.

A proof can be found here.