Background
Hey everyone, it's been a few years since college math, and I'm a bit out of practice. I'm currently working on using the exponential function to model the probability of an event happening. Right now, I've set N to 50, which means that when X hits 50, the probability of the event happening is 100%.
The function I am using right now is this:
$$ f(x)=e^{\frac{\ln\left(2\right)}{50}x}-1 $$
The graph of the function looks like this:
Question
The issue is that it is not "convex" enough for my taste. I'd like the probabilities for the earlier inputs (X=0, 1, 2, 3, ...) to be lower, while "backloading" the end – increasing the probabilities for the later inputs (X=..., 47, 48, 49, 50). Nevertheless, I'd like to maintain the same intercepts at (0, 0) and (N, 1). Could someone help me out?