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What are the general concepts and what are the instances of those concepts? Like, are probability density functions and probability mass functions instances of probability distributions?

Also, where does the study of probability begin? It seems to me that it begins with the notion of a sample space which I believe to be just a set that I, as a human being, associate to each element a possible outcome of an experiment. It basically just has to have the "right" cardinality since math doesn't really know about HEADS/TAILS or the existence of a dice.

Then an event is any subset of a sample space and a probability measure maps events to the interval [0,1] of the real numbers. But then what are probability distributions?

Random variables are functions that map from a sample space to a subset of the real numbers. They can be continuous or discrete. But the definition of continuity of functions requires the domain and image of a function to have topologies on them. Is that the case for a continuous random variable? What are the topologies on the sample space and the subset of the real numbers?