Probability: Multiplication and Addition Rules

By the end of this lesson, you will be able to apply the multiplication rule for independent and dependent events, apply the addition rule for mutually exclusive and overlapping events, and use complementary probability.

The Big Idea

Probability answers "how likely?" on a scale from 0 to 1. Two fundamental rules — the multiplication rule (for "and") and the addition rule (for "or") — handle most probability calculations. The key is classifying events correctly: are they independent or dependent? mutually exclusive or overlapping?

Vocabulary

Sample space $S$: the set of all possible outcomes.

Event $A$: a subset of outcomes.

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \quad \text{(equal likelihood)}$$

Complement: $P(A^c) = P(\text{not } A) = 1 - P(A)$

Multiplication Rule (AND)

Independent events — the outcome of one does not affect the other:

$$P(A \text{ and } B) = P(A) \cdot P(B)$$

Dependent events — the outcome of $A$ affects $B$:

$$P(A \text{ and } B) = P(A) \cdot P(B|A)$$

where $P(B|A)$ is the conditional probability of $B$ given $A$ has occurred.

Addition Rule (OR)

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Mutually exclusive events ($A$ and $B$ cannot both occur → $P(A \text{ and } B) = 0$):

$$P(A \text{ or } B) = P(A) + P(B)$$

Example 1 — Independent Events

A coin is flipped twice. Find $P(\text{two heads})$.

Flips are independent. $P(H) = 1/2$ each.

$P(HH) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Example 2 — Dependent Events (No Replacement)

Drawing 2 cards from a standard deck. Find $P(\text{both red})$.

First draw: $P(\text{red}) = 26/52 = 1/2$

Second draw (given first was red): $P(\text{red}|1\text{st red}) = 25/51$

$P(\text{both red}) = \dfrac{1}{2} \times \dfrac{25}{51} = \dfrac{25}{102}$

Example 3 — Addition Rule (Overlapping)

Rolling a die: find $P(\text{even or divisible by 3})$.

$A = {2, 4, 6}$, $P(A) = 3/6$