Compound events
When two things might happen, together or one after the other, how do their separate chances combine?
Where you are 10 inside
What this area is about
Most real questions join events: this “or” that, this “and then” that, “at least one”. Counting every outcome again works for small cases and fails quickly. This area gives the rules that build the chance of a joined event from the chances of its parts, and says exactly when each rule holds.
How these fit together
Language. Union and intersection give “or” and “and” their exact meanings, and joint and marginal probability reads them from a two-way table.
“Or” rules. Mutually exclusive events add, because they cannot happen together. The general addition rule subtracts the overlap for any two events.
“And” rules. Independent events multiply. Tree diagrams lay out stages that need not be equally likely. Dependent events, such as draws without replacement, update the chance after each stage, and the general multiplication rule P(A \text{ and } B) = P(A) \times P(B \mid A), for P(A) > 0, covers every case.
A procedure. ”At least one” is 1 minus the chance that none occur.
A sub-area. Conditional probability finds P(B \mid A) for P(A) > 0, and reverses it with Bayes’ rule.
A path through it
Union and intersection and joint probability first, then the two addition rules. Then independent events, tree diagrams, dependent events and the general multiplication rule, then “at least one”. Conditional probability comes last, since it uses nearly all of these.
Finish here and you are ready for counting and random variables.