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Probability forms the foundation for almost all treatments of statistical inference.
In our treatment, probability is a law that assigns numbers to the long run occurrence of random phenomena after repeated unrelated realizations.

Before we begin discussing probability, let's dispense with some deep philosophical questions, such as "What is randomness?" and "What is the fundamental interpretation of probability?". One could spend a lifetime studying these questions (and some have). For our purposes, randomness is any process occurring without apparent deterministic patterns. Thus we will treat many things as if they were random when, in fact they are completely deterministic. In my field, biostatistics, we often model disease outcomes as if they were random when they are the result of many mechanistic components whose aggregate behavior appears random. Probability for us will be the long run proportion of times some occurs in repeated unrelated realizations. So, think of the proportion of times that you get a head when flipping a coin.
Before we begin discussing probability, let's dispense with some deep philosophical questions, such as "What is randomness?" and "What is the fundamental interpretation of probability?". One could spend a lifetime studying these questions (and some have). For our purposes, randomness is any process occurring without apparent deterministic patterns. Thus we will treat many things as if they were random when, in fact they are completely deterministic. In my field, biostatistics, we often model disease outcomes as if they were random when they are the result of many mechanistic components whose aggregate behavior appears random. Probability for us will be the long run proportion of times some event occurs in repeated unrelated realizations. So, think of the proportion of times that you get a head when flipping a coin.

For the interested student, I would recommend the books and work by Ian Hacking to learn more about these deep philosophical issues. For us data scientists, the above definitions will work fine.

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