The Counting Principle
Master the Fundamental Counting Principle, factorials, arrangements with and without repetition, and constraint problems — with the slot method and fully worked examples.
Probability
Table of Contents
Probability: The Logic of Counting (~15 marks, Paper 1)#
Probability is worth ~15 marks in Paper 1. In Grade 12, the Counting Principle and factorials are the main new content, but you must also be solid on the probability rules from Grade 10–11.
The Fundamental Counting Principle#
If you have 3 shirts and 4 pants, how many outfits can you make?
- You have 3 choices for slot 1.
- You have 4 choices for slot 2.
- Multiply them! $3 \times 4 = 12$.
The general rule: If there are $n_1$ ways to do task 1, $n_2$ ways to do task 2, …, then the total number of ways = $n_1 \times n_2 \times \ldots$
Factorials#
$$n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1$$| $n$ | $n!$ | Meaning |
|---|---|---|
| $1$ | $1$ | |
| $2$ | $2$ | |
| $3$ | $6$ | |
| $4$ | $24$ | |
| $5$ | $120$ | Ways to arrange 5 objects |
| $6$ | $720$ | |
| $0$ | $1$ | By definition (the empty arrangement) |
⚠️ $0! = 1$, not $0$. This is a definition that makes the formulas work correctly.
Common Counting Scenarios#
| Scenario | Formula | Example |
|---|---|---|
| Arrange $n$ objects in a row | $n!$ | 5 people in a line: $5! = 120$ |
| Arrange with some identical | $\frac{n!}{k_1! \cdot k_2! \cdot \ldots}$ | Letters of PEPPER: $\frac{6!}{3! \cdot 1! \cdot 2!} = 60$ |
| Choose $r$ from $n$ (order matters) | $\frac{n!}{(n-r)!}$ | Pick 3 from 7 in order: $\frac{7!}{4!} = 210$ |
| Constraints (e.g., must start with…) | Fix the constrained slot, count the rest | 4-digit code starting with 5: $1 \times 10 \times 10 \times 10 = 1000$ |
Probability Rules Revision (Grade 10–11)#
| Rule | Formula | Use when… |
|---|---|---|
| Addition (OR) | $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ | Finding $P$(at least one event) |
| Product (AND) | $P(A \text{ and } B) = P(A) \times P(B | A)$ |
| Independent events | $P(A \text{ and } B) = P(A) \times P(B)$ | Events don’t affect each other |
| Complementary | $P(A') = 1 - P(A)$ | “At least one” problems |
| Mutually exclusive | $P(A \text{ and } B) = 0$ | Events can’t happen together |
Deep Dives (click into each)#
- Probability Rules & Identities — addition rule, mutually exclusive, independent events, complementary events, Venn diagrams, and tree diagrams
- The Counting Principle & Factorials — slot method, arrangements, repetitions, and constraint problems
🚨 Common Mistakes#
- Forgetting $0! = 1$: This comes up in formulas. It’s a definition, not a calculation.
- Counting with vs without repetition: “Digits may repeat” → multiply by 10 each time. “No repeats” → decreasing choices ($10 \times 9 \times 8 \times \ldots$).
- Not identifying constraints first: If a problem says “must start with a vowel”, fill the constrained slot FIRST, then count the remaining slots.
- Confusing independent and mutually exclusive: Mutually exclusive = $P(A \text{ and } B) = 0$. Independent = $P(A \text{ and } B) = P(A) \times P(B)$. They are NOT the same thing.
- Identical objects: When arranging letters with repeats (like MISSISSIPPI), you MUST divide by the factorial of each repeated letter.
🔗 Related topics:
- Statistics — probability and statistics are complementary
- Sequences & Series — factorial notation appears in series formulas
📌 Grade 11 foundation: Probability: Combined Events — tree diagrams, independence tests, contingency tables
⏮️ Differential Calculus | 🏠 Back to Grade 12 | ⏭️ Trigonometry
Probability Rules & Identities
Master the addition rule, mutually exclusive events, independent events, and complementary events — the foundation for all probability calculations.