Nothing in this chapter is new mathematics. Every idea here — set operations, the product and sum rules, permutations and combinations, Pascal's triangle, sigma notation, geometric series, improper integrals, limits at infinity — was taught in an earlier course. What's new is the question they're all about to answer together: given an experiment whose outcome isn't known in advance, how likely is a given result?
That question turns out to need every one of these tools at once. Events are described with set operations. Sample spaces are sized with the product and sum rules. The named distributions ahead are built from permutations, combinations, and Pascal's triangle. A pmf's validity is checked with a sigma sum; a geometric distribution's validity, with a geometric series. A continuous density's validity leans on definite and improper integrals, and a continuous distribution's tail behavior leans on limits at infinity.
This chapter doesn't re-teach any of that from scratch — it refreshes each tool quickly, in exactly the framing probability will use it, so Chapter 1 can start immediately from a shared, sharpened foundation instead of pausing to relearn algebra mid-argument.
By the end, every section from here through the end of the course will be able to assume this toolkit is already in your hands.