Probability
How do you put a number on how likely something is, and what can you do with that number once you have it?
Where you are 8 inside
What this area is about
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). It starts with single events, combines them, counts outcomes too many to list, and then attaches numbers to outcomes so that a whole distribution has a center and a spread.
How these fit together
Foundations. Basic probability finds a chance from a model (counting equally likely outcomes, or a region) or from repeated trials. Compound events give the rules for “or”, “and” and “at least one”, and hold conditional probability, defined when the event conditioned on has probability above 0.
A tool. Counting finds how many ways something can happen without listing them: permutations, combinations, the binomial theorem.
Distributions. Random variables attach a number to each outcome and describe its mean and spread, including the binomial and geometric distributions.
College level, not yet written. Foundations: probability from axioms, counting models and conditioning. Random variables: discrete and continuous distributions, expectation and variance. Several random variables: joint distributions, conditional expectation and moment generating functions. Limit theorems: why averages settle down, and why they look normal.
A path through it
Basic probability and compound events go together: start with outcomes, events and counting outcomes, take compound events, then return for the law of large numbers and simulation. Counting comes next, then random variables. At college level the order is foundations, random variables, several random variables, then limit theorems.
After the school-level areas, density curves and sampling distributions carry this into statistics.