Probability and Counting Principles

$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

$0 \leq P \leq 1$. $P(\text{not A}) = 1 - P(A)$.

Compound Events

• AND (independent): $P(A \text{ and } B) = P(A) \times P(B)$.
• OR (mutually exclusive): $P(A \text{ or } B) = P(A) + P(B)$.
• OR (not mutually exclusive): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Fundamental Counting Principle

If one event has $m$ outcomes and another has $n$ outcomes, total outcomes = $m \times n$.

Permutations (Order Matters)

$P(n, r) = \frac{n!}{(n-r)!}$

Combinations (Order Doesn't Matter)

$C(n, r) = \frac{n!}{r!(n-r)!}$

• Does order matter? → Permutation. Arranging books on a shelf.
• Order doesn't matter? → Combination. Choosing a committee.