## A note from myself January 14, 2010

Posted by eric22222 in General, Math.

This morning, I woke up to find a sticky note next to my alarm clock. The text was sloppy, but it was definitely my handwriting. I didn’t fully remember writing anything out, but there it was: some simple equation.

$\lim \limits_{p \to \infty} 1 - (1-p)^{p^{-1}}$

I have some vague memories of thinking through some stuff last night, but nothing too concrete. Okay, so p must be some probability (I always use p for that), so one minus p is the chance of something not happening. That raised to the inverse of p… that’d be the chance of something not happening a number of times in a row. One minus that value. That’d be the chance of all that not happening.

Aha! So if p were the chance of, say, getting heads on a coin flip (50%), $1 - (1-p)^{p^{-1}}$ is the chance of not getting tails twice in a row. That is, the chance that heads will come up within the first two tosses. Seventy-five percent.

But why the limit?

Well, if p were only one-third… Ah, our chance of it cropping up in the first three times is 19/27, or roughly seventy percent. If we drop p to one-fourth, our chance goes down to sixty-eight percent. Oh, I see… it’s approaching some value, but what?

p, $1 - (1-p)^{p^{-1}}$
1, 1.000
2, 0.750
3, 0.704
4, 0.684
5, 0.672
6, 0.665

What in the world is this thing approaching? This doesn’t look like anything I’m familiar with…

$10^2$, 0.633967659
$10^3$, 0.632304575
$10^4$, 0.632138954
$10^5$, 0.632122398
$10^6$, 0.632120743
$10^7$, 0.632120577

It’s a limit, so it should have something to do with e (2.71828), but I don’t see how- oh. Wait, now I do. This value we’re approaching is $1 - \frac{1}{e}$. So the chance of success of a task of probability p within the first $p^{-1}$ trials approaches one minus the inverse of e. Hope that’s what you were looking for, past self.