In Grade 10, you worked with Venn diagrams and the addition rule for single events. In Grade 11, we ask: “What happens when events happen in sequence?” Does the first event change the second? This leads to the crucial distinction between independent and dependent events.
| Rule | Formula | Use when… |
|---|---|---|
| Addition rule (OR) | $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ | Finding the probability of at least one event |
| Product rule (AND) | $P(A \text{ and } B) = P(A) \times P(B\|A)$ | Finding the probability of both events |
For independent events, the product rule simplifies to:
$$P(A \text{ and } B) = P(A) \times P(B)$$| Independent | Dependent | |
|---|---|---|
| Definition | One event does NOT affect the other | One event CHANGES the other’s probability |
| Example | Flipping a coin twice | Drawing 2 cards without replacement |
| The test | $P(A \text{ and } B) = P(A) \times P(B)$ | $P(A \text{ and } B) \neq P(A) \times P(B)$ |
| Product rule | $P(A) \times P(B)$ | $P(A) \times P(B\|A)$ |
⚠️ Independent ≠ Mutually exclusive! If events are mutually exclusive ($P(A \text{ and } B) = 0$), they are actually very dependent — if one happens, the other definitely cannot.
Tree diagrams map out every possible outcome in a multi-step experiment.
Rules:
Example: A bag has 3 red and 2 blue balls. Two are drawn without replacement.
| Draw 1 | Draw 2 | Probability |
|---|---|---|
| R then R | $\frac{3}{5} \times \frac{2}{4}$ | $= \frac{6}{20}$ |
| R then B | $\frac{3}{5} \times \frac{2}{4}$ | $= \frac{6}{20}$ |
| B then R | $\frac{2}{5} \times \frac{3}{4}$ | $= \frac{6}{20}$ |
| B then B | $\frac{2}{5} \times \frac{1}{4}$ | $= \frac{2}{20}$ |
| Total | $= \frac{20}{20} = 1$ ✓ |
💡 Without replacement: The denominator drops by 1 on the second draw (and the numerator changes too if you drew one of that colour). This is what makes it dependent.
A contingency table organises data by two categories simultaneously:
| Likes Maths | Doesn’t Like Maths | Total | |
|---|---|---|---|
| Male | 30 | 20 | 50 |
| Female | 25 | 25 | 50 |
| Total | 55 | 45 | 100 |
Testing for independence: Events are independent if every cell equals:
$$\frac{\text{row total} \times \text{column total}}{\text{grand total}}$$If even ONE cell doesn’t match, the events are dependent.
🔗 Related Grade 11 topics:
- Statistics: Standard Deviation — contingency tables bridge statistics and probability
- Quadratic Equations — probability equations sometimes require algebraic solving
📌 Grade 10 foundation: Probability Basics and Venn Diagrams
📌 Grade 12 extension: Probability & Counting Principle — factorials, permutations, combinations
⏮️ Finance, Growth & Decay | 🏠 Back to Grade 11 | ⏭️ Trigonometry
