🎲 Basic Probability Rules

Fundamental Definitions

Sample Space ($S$)

• Set of all possible outcomes
• Example: rolling die → $S = \{1, 2, 3, 4, 5, 6\}$

Event ($A$)

• Subset of sample space
• Example: $A =$ "roll even" $= \{2, 4, 6\}$

Probability

• P(A) = \frac{\text{# favorable outcomes}}{\text{# total outcomes}} (assuming equally likely)
• Ranges from 0 (impossible) to 1 (certain)

The Complement Rule

Definition: $A^c$ is the complement (event does NOT occur)

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

Example: If $P(\text{pass}) = 0.7$, then $P(\text{fail}) = 1 - 0.7 = 0.3$

Use case: Sometimes easier to calculate $P(A^c)$ directly

Mutually Exclusive Events

Definition: Events cannot occur simultaneously; $P(A \cap B) = 0$

Example: Single die roll → "even" and "odd" are mutually exclusive

Implication: If $A$ and $B$ mutually exclusive, then $P(A \cap B) = 0$

Addition Rule for Mutually Exclusive Events

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

Example: $P(\text{roll 1 or 2}) = P(\text{roll 1}) + P(\text{roll 2}) = 1/6 + 1/6 = 1/3$

General Addition Rule (for Any Events)

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

Subtract $P(A \cap B)$ to avoid double-counting

Example:

• $P(A) = 0.6$, $P(B) = 0.4$, $P(A \cap B) = 0.1$
• $P(A \cup B) = 0.6 + 0.4 - 0.1 = 0.9$

Worked Example

Scenario: Polling 1000 voters

• 600 support tax increase
• 400 oppose tax increase
• 100 of supporters also want education reform

What's the probability a voter supports tax increase OR education reform?

$P(\text{tax} \cup \text{reform}) = P(\text{tax}) + P(\text{reform}) - P(\text{tax} \cap \text{reform})$ $= 0.6 + ? - 0.1$

Need $P(\text{reform})$—not given! Use Venn diagram or two-way table.

Common Mistakes

• Using addition rule for non-mutually-exclusive events without subtracting overlap
• Confusing "and" ($\cap$) with "or" ($\cup$)
• Forgetting to subtract $P(A \cap B)$ in general rule

Decision Rule

Are events mutually exclusive? → Use $P(A) + P(B)$ Can they overlap? → Use $P(A) + P(B) - P(A \cap B)$

AP Exam Tip

Draw Venn diagrams or two-way tables for clarity. The general addition rule always works—apply it first when unsure.