Those are the most basic concepts of probability theory, but how they are related / linked is rarely discussed in textbooks (at least those that I read).
An event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. (from https://en.wikipedia.org/wiki/Event_(probability_theory))
A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. (from http://www.stat.yale.edu/Courses/1997-98/101/ranvar.htm)
Here are my thoughts. Would you agree? Any suggestions / references are very welcomed.
Consider as an example rolling dice twice. Random variable X is the sum of two rollings. So, for example, X({3,2}) = 5. I can say that {3,2} is my event that occured. In other words, random variable is a function that gets as input the specific event that occured. So, the input of random variable is a special case of event with only one outcome. The value of random variable (i.e., the output of random variable) will be associated with this specific event. Obviously, the same value of random variable can be associated with many events (e.g., X({3,2}) = 5, X({4,1}) = 5).
We can also consider probability of the random variable vs. probability of the event. Let’s say my event is A = {6,6}. So, P(X>11) = P(A). So, I can say that: “probability of random variable X to be more than 11 is equal to probability of event A (getting twice 6)”. In this example, the A contained only one potential outcome. But if I put it: P(X>5) = P(B), then B will contain many potential outcomes.
Thanks!