Probability Quick Reference Card
Module 3: Probability Basics
GOLDEN RULES: 0 ≤ P(A) ≤ 1 • P(S) = 1 • P(not A) = 1 − P(A)
Basic Probability
P(A) = n(A) / n(S)
Favorable outcomes / Total outcomes
P(not A) = 1 − P(A)
Complement Rule
Counting Methods
FCP: n₁ × n₂ × n₃ × ...
Factorial: n! = n × (n−1) × ... × 1
Permutation: P(n,r) = n!/(n−r)!
(Order matters)
Combination: C(n,r) = n!/(r!(n−r)!)
(Order doesn't matter)
Swap items = different? YES → Perm, NO → Comb
Conditional Probability
P(A|B) = P(A and B) / P(B)
P(A and B) = P(A) × P(B|A)
Independence Test
A and B independent if:
- P(A|B) = P(A), OR
- P(A and B) = P(A) × P(B)
Probability Distributions
Valid Distribution
- 0 ≤ P(X) ≤ 1 for all X
- ΣP(X) = 1
E(X) = Σ[x · P(x)]
Expected Value (mean)
Var(X) = Σ[(x−μ)² · P(x)]
SD(X) = √Var(X)
Common Examples
| Situation | Method |
|---|---|
| Coin flip | P(H) = 1/2 |
| Die roll | P(any) = 1/6 |
| Card from deck | P(suit) = 13/52 |
| Arranging books | n! |
| Committee selection | C(n,r) |
| Race winners | P(n,r) |
Key Decision Tree
Counting Problem?
- Multiple stages? → FCP
- Arrange all? → n!
- Order matters? → P(n,r)
- Order doesn't? → C(n,r)
Probability Problem?
- Basic? → n(A)/n(S)
- "Not"? → 1 − P(A)
- "Given"? → P(A|B)
- "And"? → P(A) × P(B|A)
- Independent? → P(A) × P(B)
Critical Reminders
- P(A) always between 0 and 1
- All P(X) in distribution sum to 1
- E(X) = long-run average
- WITHOUT replacement → dependent
- WITH replacement → independent
- Order matters → permutation
- Order doesn't → combination
- P(not A) = 1 − P(A)
Free Statistics Learning Platform • Safaa Dabagh • sdabagh.github.io • © 2025