Math Q: probability function and related formulas

Factual Questions
1

This is probably a fairly basic question but unfortunately my math education was sadly neglected. I was curious about the following:

Let’s say that tickets for a raffle have one chance in a hundred of winning, and you buy a hundred tickets. If you have one chance in N of something happening, and you have N trials, then the chance of it NOT happening is (N-1/N)N. What is the limit of this function when N approaches infinity? For large values of N it seems to hover around .36; is there some mathematical significance to this value? And finally, if you graphed N and the outcome of (N-1/N)N, you’d get a curve that I presume would be parabolic or hyperbolic. Would the Cartesian formula for this curve relate to the definition of the function in some way?

2

That should be (N-1/N)[sup]N[/sup].

3

I think it should actually be ((N-1)/N)[sup]N[/sup] - which is equal to (1-1/N)[sup]N[/sup]

4

The limit is 1/e (which can be derived by replacing N with M+1), and the curve isn’t parabolic or hyperbolic.

5

Is it not closer to .368, the reciprocal of 2.71828 ?

That number is Euler’s number e (not to be confused with the unrelated “Euler’s constant”).

6

This.

7

So what sort of curve is it, and can you express it in terms of x and y?

8

If I understand what you mean, of course you can: just write it with x instead of N as the independent variable.

Like this.

9

The limit, as N approaches infinity, of (1 + x/N)[sup]N[/sup] is e[sup]x[/sup]. The OP’s formula gives the case where x = -1.

10

One with a horizontal asymptote, I gather.

11

Ah, I had only considered the curve for values of N>=2.

If I win the lottery I’m going to take courses in all the trig and calc I never got around to.