Basic Probability Rules

Sample Spaces and Events

• Sample space $S$: The set of all possible outcomes
• Event: A subset of the sample space

Probability Axioms

For any event $A$:

1. $0 \leq P(A) \leq 1$
2. $P(S) = 1$
3. For mutually exclusive events $A$ and $B$: $P(A \cup B) = P(A) + P(B)$

Complement Rule

$P(A^c) = 1 - P(A)$

The probability that event $A$ does not occur equals 1 minus the probability that it does.

Addition Rule (General)

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

This accounts for double-counting when events overlap.

Special case — Mutually Exclusive Events ($A \cap B = \emptyset$): $P(A \cup B) = P(A) + P(B)$

Multiplication Rule (General)

$P(A \cap B) = P(A) \cdot P(B|A)$

Special case — Independent Events: $P(A \cap B) = P(A) \cdot P(B)$

Mutually Exclusive vs. Independent

| Property | Mutually Exclusive | Independent | |----------|-------------------|-------------| | Definition | Cannot occur together | Occurrence of one doesn't affect the other | | Formula | $P(A \cap B) = 0$ | $P(A \cap B) = P(A) \cdot P(B)$ | | Can both be true? | Only if $P(A) = 0$ or $P(B) = 0$ | |

Important: If two events have non-zero probabilities and are mutually exclusive, they cannot be independent (and vice versa).

Probability Models

A probability model lists all outcomes and their probabilities:

• Every probability is between 0 and 1
• All probabilities sum to 1

AP Tip: The most common mistake is confusing "or" (addition rule) with "and" (multiplication rule). "Or" → add; "And" → multiply.