# Probability of X>Y based on single observation?

I'm struggling a bit with a problem -more specifically on how to tackle the problem- and came for help or pointers. I'm an engineer with very little education in statistics but I'm really motivated in getting a better understanding of the resolution of such problems.

I have two independent players playing darts and I would like to express the probability that player 1 is better than player 2 based on a single measurement which consists of the sum of N throws of each player. For the sake of simplicity, I assumed that the outcome of a measurement depends only on one single parameter and I can already simulate without problem the density function of a N-throws based on that parameter. Intuitively, I already see that players having the same parameter value can generate measurements that are, on average, 50% better than the other player (one wins a match, then the other wins it etc.).

What I would like to do now is to answer the question about the probability of player 1 being better than player 2, or conversely, the probability that they are equally good based on a single match result.

How should I approach this problem ? I'm not necessarily looking for an analytical form as long as I can find a method to populate a table giving confidence in the assertion based on the two scores that can only take bounded values between 0 and 10*N.

A simple way to proceed is to do an hypothesis sign-test that is a binomial test. $$H_0: p=0.5$$ against $$H_1: p>0.5$$
You reject $$H_0$$ if your fixed p-value is less than a fixed %
For example, if you have $$N=10$$ throwns with the following winners
$$\{x,x,y,x,x,y,x,x,x,x\}$$
You set 1 if x and 0 if Y thus your result is 8 and being the probability that a $$Bin(10;0,5)\geq 8=0.0547$$ you can reject the hypothesis that $$X\leq Y$$ at any significance level greater than 5.47%