In the formulas below, we will presume that we have a random variable 5.2.1 \(X\) which maps the sample space S onto some range of real numbers \(R\text{.}\) From this set, we then can define a probability function \(f(x)\) which acts on the numerical values in \(R\) and returns another real number. We attempt to do so to obtain (for discrete values) P(sample space value s)\(= f(X(s))\text{.}\) That is, the probability of a given outcome s is equal to the composition which takes s to a numerical value x which is then plugged into f to get the same final values.