Confusion on the definition of variance on a random variable

Question

For $X(s)$, I interpret that as the value the random variable gives based on the input of an element in the sample space. $E(X)$ is the sort of mean for all of the expected values in the sample space. $X(s) - E(X)$ is being subtracted to find the distance from the value of $X(s)$ to the average value of all $X(s)$. It is then squared to account for negative distances. All of this is summed to calculate the variance of all the sample space of random variables rather than just one instance. However, I'm confused as to what the $p(s)$ does in the formula.

Answer 1

$p(s)$ is the individual probability for it's corresponding value of $X(s).$ Its the random variable version of dividing by the sample space in the formula of population / sample variance