The Addition and Multiplication Rules

The addition and multiplication rules are simply how you combine probabilities: "or" questions add, and "and" questions multiply. The symbols look intimidating, but $\cup$ just means "or" (at least one happens) and $\cap$ means "and" (both happen). The addition rule is $P(A) + P(B)$ (count both events) minus $P(A \cap B)$ (the overlap, which got counted twice). The multiplication rule is $P(A)$ (A happens) times $P(B \mid A)$ (B happens too, given that A did); the bar $\mid$ is read "given." Two rules, a few symbols, and you've got it.

Why subtract the overlap? Picture a class where 40% play a sport and 50% play an instrument. If you add 40% + 50%, every student who does both got counted once in each group. Subtracting $P(A \cap B)$ once fixes the double count.

Why multiply? Think of it as "a fraction of a fraction." If A happens 40% of the time, and B happens in half of those cases, then both happen in half of 40%, which is 20%. That second fraction is the conditional probability $P(B \mid A)$, the chance of B once you're only looking at cases where A happened. There's a picture of this further down, where probabilities are areas in a square.

Mutually exclusive events can't happen together: $P(A \cap B) = 0$, so $P(A \cup B) = P(A) + P(B)$.

Independent events don't affect each other: $P(B \mid A) = P(B)$, so $P(A \cap B) = P(A)P(B)$.

Without replacement the probabilities change after each draw. Two aces in a row from a deck: $\frac{4}{52}\cdot\frac{3}{51}$. After the first ace, only 3 aces are left among 51 cards, and that's exactly the conditional probability.

A common mistake. "Mutually exclusive" and "independent" both sound like "the events have nothing to do with each other," so it's tempting to treat them as the same thing. They're nearly opposites. Mutually exclusive events with positive probabilities are never independent: if one happens, you know for sure the other didn't, so knowing one tells you a lot about the other.

Reliability engineers multiply component success probabilities to rate a system; spam filters and recommendation engines are built on conditional probabilities.