Basic Probability Rules

Probability Basics

Probability: Measure of likelihood an event occurs (0 to 1)

P(A) = 0 → Event A is impossible
P(A) = 1 → Event A is certain
0 < P(A) < 1 → Event A may or may not occur

Complement: Event A doesn't occur, denoted $A^c$ or $\bar{A}$

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

Sample Space and Events

Sample Space (S): Set of all possible outcomes
Event: Subset of sample space

Example: Roll a die

• Sample space: S = {1, 2, 3, 4, 5, 6}
• Event "even number": E = {2, 4, 6}
• P(E) = 3/6 = 0.5

Computing Probability

Equally likely outcomes:

$P(A) = \frac{\text{Number of outcomes in A}}{\text{Total number of outcomes}}$

Example: Deck of cards, P(Heart) = 13/52 = 1/4

Relative frequency (empirical probability):

$P(A) \approx \frac{\text{Number of times A occurred}}{\text{Total number of trials}}$

Addition Rule (OR)

For any two events A and B:

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

Why subtract P(A ∩ B)? Avoid double-counting outcomes in both A and B

Example: Draw one card

• P(Heart) = 13/52
• P(Face card) = 12/52
• P(Heart and Face) = 3/52
• P(Heart or Face) = 13/52 + 12/52 - 3/52 = 22/52

Mutually Exclusive Events

Definition: Events that cannot both occur (no overlap)

If A and B are mutually exclusive: P(A ∩ B) = 0

Addition Rule simplifies:

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

Example: Roll a die

• Event A: Roll 2
• Event B: Roll 5
• These are mutually exclusive (can't roll both)
• P(A or B) = 1/6 + 1/6 = 2/6 = 1/3

Non-example: P(Heart) and P(Ace) are NOT mutually exclusive (Ace of Hearts is in both)

Multiplication Rule (AND)

For any two events:

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

Where P(B|A) = probability of B given A occurred

We'll explore this more in conditional probability topic

Independent Events (Preview)

If A and B are independent: P(B|A) = P(B)

Multiplication Rule simplifies:

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

Example: Flip coin twice

• P(First heads) = 1/2
• P(Second heads) = 1/2
• P(Both heads) = 1/2 × 1/2 = 1/4

Probability Rules Summary

Rule 1: For any event A, 0 ≤ P(A) ≤ 1

Rule 2: P(S) = 1 (something in sample space must occur)

Rule 3: Complement Rule: P(A^c) = 1 - P(A)

Rule 4: Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)

Rule 5: If mutually exclusive: P(A or B) = P(A) + P(B)

Rule 6: Multiplication Rule: P(A and B) = P(A) × P(B|A)

Rule 7: If independent: P(A and B) = P(A) × P(B)

Venn Diagrams

Visual tool for probability:

Union (A or B): Everything in A or B or both
Intersection (A and B): Overlap of A and B
Complement (A^c): Everything outside A

Use Venn diagrams to visualize and organize probability problems.

Tree Diagrams

Useful for sequential events:

• Each branch represents outcome
• Multiply probabilities along path
• Add probabilities of different paths to same outcome

Example: Flip coin twice

• First flip: 1/2 Heads, 1/2 Tails
• Second flip: 1/2 Heads, 1/2 Tails
• P(HH) = 1/2 × 1/2 = 1/4
• P(exactly one head) = P(HT) + P(TH) = 1/4 + 1/4 = 1/2

Common Mistakes

❌ Adding probabilities when should multiply (AND vs OR confusion)
❌ Forgetting to subtract overlap in addition rule
❌ Assuming events are independent when they're not
❌ Confusing mutually exclusive with independent

Practice Strategy

1. Identify: What event(s) are we finding probability for?
2. Determine: AND (multiply) or OR (add)?
3. Check: Mutually exclusive? Independent?
4. Calculate: Apply appropriate rule
5. Verify: Does answer make sense (between 0 and 1)?

Quick Reference

Complement: P(A^c) = 1 - P(A)
OR (Addition): P(A or B) = P(A) + P(B) - P(A and B)
AND (Multiplication): P(A and B) = P(A) × P(B|A)
Mutually Exclusive OR: P(A or B) = P(A) + P(B)
Independent AND: P(A and B) = P(A) × P(B)