Thread:Homuhomu123/@comment-24921408-20150202200415/@comment-24921408-20150304232420

Hey Homuhomu!

I've taken a look at the idea behind the standard error calculation, and I understand it better now. It's, as you have mentioned, great for any probabilities around 50%, which would work pretty well for most combat calculations I think. The error bounds are symmetrical, and they're kind of fixed, because they assume that the probability is 50%, which is when the largest error interval results. Another way of putting it is that it errs on the side of caution. So if you are at around the tail ends with high or low probabilities, the error estimated is going to be significantly larger than what it should be (causing you to work harder to achieve a narrower bound), which, although isn't necessarily a bad thing, as the work is already tedious as it is, it's best to know how how few samples are needed to ensure enough confidence.

As also mentioned Mathiaszealot, "the proper method is to use the integral of a beta function", which I believe what I've managed to work out essentially does (it uses the Incomplete Beta Function which is a more general form of the beta function). I also realised that there's one more example in the Checking Whether a Coin is Fair wikipedia article which I tested against. What the example does is to use the standard error calculation to generate the bounds of the confidence interval for a 12000 coin toss with 5961 heads, and assumes that the underlying probability is 5961/12000 = 49.68% (which is close to 50%). To verify, I plugged the same figures into the MATLAB script, and it generated the same intervals. So this basically shows that the standard error calculation is valid for probabilities around 50%, which is what we already know and realise (just that now, it's been verified).

So I suppose, if you need something quick, you could use the table you generated, since it's much easier to refer to. However, if you want a more thorough calculation, or if you're at the computer and have access to the MATLAB script, it would be better to plug the values in and let the computer churn out the exact confidence bounds.