# Calculate probability when comparing probability distributions

I'm essentially trying to find a general way to predict the probability of one outcome when comparing two functions that are described as bell curves or probability distributions.

The specific example I am using is writing a code comparing different xky probability distributions for Legend of the Five rings. And using that to say what the odds one or the other win in a contested roll. I'm bringing this here rather than stackoverflow as I feel the problem is with my approach rather than the code.

What I have done so far is to treat the bell curve as a probability curve by just sampling it and then saying the probability that character no. 1 loses to character 2 is equal to the odds that they roll below one of my sampling points multiplied by the odds no.2 rolls above it.

However while I can treat the rolls of no 1. and 2 as independent events I cannot say the the probability of no.1 rolling under a value of 5 and 10, however I don't see how to properly correct for this.

I don't believe that convolution will give me the correct answer as I'm not looking to see how similar they are.

If the two rolls have the distribution of a normal curve, possibly with different means and variances ($\sigma^2$) then the distribution of the difference is a normal curve with mean equal to the difference in means, and variance equal to the sum of the variances. So you immediately know how many $\sigma$ the mean is below or above zero, and the normal curve has some area more than than many $\sigma$ above or below the mean.