what can we get if we add up the values that each random variable was assigned, along with the appropriate probabilities?

You’re describing the expected value (mean) of a random variable.

When you “add up the values that each random variable was assigned, along with the appropriate probabilities,” you are forming a weighted average of the possible values of the random variable, where each value is weighted by its probability.

Formally, for a discrete random variable XXX that can take values x1,x2,…x_1,x_2,\dots x1​,x2​,… with probabilities P(X=xi)P(X=x_i)P(X=xi​), the quantity you get is

E(X)=∑ixi P(X=xi),E(X)=\sum_i x_i,P(X=x_i),E(X)=i∑​xi​P(X=xi​),

which is called the expected value or mean of XXX.

Intuitively, this is the long‑run average outcome you would see if you repeated the random experiment many times.