What is a Random Variable?
A random variable is a function that assigns a number to each outcome. Nothing more mysterious than that, and the name is unhelpful: it is not random and it is not a variable, it is a mapping.
Flip a coin and the outcomes are heads and tails. Define for heads and for tails, and now you have something you can average, add to other quantities and put a price on. That is the entire point. Probability describes what might happen; random variables make it arithmetic.
Why the mapping is a modelling choice
The same experiment supports many random variables, and choosing the right one often is the problem.
Roll two dice. You could define as the sum, the maximum, the number of sixes, or the absolute difference. Each is a legitimate random variable over the same sample space, each has a different distribution, and a question about "the chance of at least one six" becomes easy or hard depending on which you pick.
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