In the bullet point definition, > the function which maps sample space outcomes to numbers, the "numbers" referred to in the codomain are not probabilities. S

In the bullet point definition, > the function which maps sample space outcomes to numbers, the "numbers" referred to in the codomain are not probabilities. Similarly, in the Wikipedia definition, > a measurable function from a sample space... to a measurable space, the "measurable space" codomain has nothing to do with probability of events. Random variables are related to probability distributions/measures, but they are not the same kind of mathematical object. As a concrete example, consider gambling on a fair coin flip. We can bet 1 unit of money on Heads and define a random variable for the amount of money we win/lose. First we set up the probability space. The sample space is the 2-element set of coin sides, Ω = {H, T} (Heads and Tails). The event set (aka sigma-algebra) F is the power set of Ω: F = {{}, {H}, {T}, {H, T}}. The probability measure assigns probability 0 to the event {}, probability 0.5 to each of the events {H}, {T}, and probability 1 to the event {H, T}. Now, we can define the random variable of our winnings by the function X mapping Ω to the real numbers by Χ(ω) = 1 if ω = H, -1 if ω = T. So, within the setting of the probability space (Ω, F, P), the random variable is defined by the function X, which does not map anything to a probability. It maps coin sides to real numbers, including negative ones! We have not yet constructed any object that gives the probabilities of the possible values of X. The only thing that "maps an outcome to a probability" in our setup so far is the measure P, but that is defined independently of our numeric random variable. If we want to talk about the probabilities that X takes the values -1 and 1, we are talking about the pushforward measure of P by X. This is another mathematical object, distinct from the random variable X itself. This is one of the points of the article. Suppose we define another random variable Y for the winnings of the person we are gambling against: Y(ω) = -1 if ω = H, 1 if ω = T. Now X and Y are two different random variables, but they have the same distributions! The pushforward measures of P by X and Y both put probability 0.5 on the measurable sets {-1} and {1}.