The Key Symbols#
| Symbol | Meaning | Keywords |
|---|---|---|
| $A \cap B$ | Intersection — “AND” | Both, overlap |
| $A \cup B$ | Union — “OR” | Either, at least one |
| $A'$ | Complement — “NOT A” | Everything except A |
| $S$ | Sample space | Everything |
1. The Addition Rule#
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$Why subtract? When you add $P(A)$ and $P(B)$, you count the overlap TWICE. Subtract it once to correct.
Worked Example#
In a class of 30 learners, 18 play soccer, 12 play cricket, and 7 play both.
$P(\text{soccer or cricket}) = \frac{18}{30} + \frac{12}{30} - \frac{7}{30} = \frac{23}{30}$
$P(\text{neither}) = 1 - \frac{23}{30} = \frac{7}{30}$
2. Filling in a Venn Diagram — The Method#
Always start from the INSIDE out:
- Fill in the intersection ($A \cap B$) first.
- Fill in the rest of A = $n(A) - n(A \cap B)$.
- Fill in the rest of B = $n(B) - n(A \cap B)$.
- Fill in the outside = $n(S) - n(A \cup B)$.
Worked Example#
$n(S) = 50$, $n(A) = 25$, $n(B) = 20$, $n(A \cap B) = 8$.
| Region | Calculation | Value |
|---|---|---|
| $A \cap B$ (overlap) | Given | 8 |
| A only | $25 - 8$ | 17 |
| B only | $20 - 8$ | 12 |
| Neither | $50 - (17 + 8 + 12)$ | 13 |
Check: $17 + 8 + 12 + 13 = 50$ ✓
Reading Probabilities from the Diagram#
$P(A \text{ only}) = \frac{17}{50}$
$P(A \cup B) = \frac{17 + 8 + 12}{50} = \frac{37}{50}$
$P(A') = \frac{12 + 13}{50} = \frac{25}{50} = \frac{1}{2}$
3. Mutually Exclusive Events#
Two events are mutually exclusive if they CANNOT happen at the same time:
$$ P(A \cap B) = 0 $$The Venn diagram shows two circles that don’t overlap.
The addition rule simplifies to: $P(A \cup B) = P(A) + P(B)$
Worked Example#
A die is rolled. Let A = {getting a 2} and B = {getting a 5}.
$P(A \cap B) = 0$ (you can’t roll both at once)
$P(A \cup B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$
4. Complementary Events#
$$ P(A') = 1 - P(A) $$A and A’ are always mutually exclusive, and together they cover the entire sample space.
Worked Example#
$P(\text{passing}) = 0.72$
$P(\text{failing}) = 1 - 0.72 = 0.28$
5. Three-Set Venn Diagrams (Extension)#
With three events A, B, C, you have 8 regions. Fill from the innermost region out:
- $A \cap B \cap C$ (all three)
- Each pairwise intersection minus the triple
- Each set alone minus all overlaps
- Outside all three
Worked Example#
In a survey of 100 people: 40 like tea, 35 like coffee, 25 like juice, 15 like tea and coffee, 10 like tea and juice, 8 like coffee and juice, 5 like all three.
| Region | Value |
|---|---|
| All three | 5 |
| Tea & Coffee only | $15 - 5 = 10$ |
| Tea & Juice only | $10 - 5 = 5$ |
| Coffee & Juice only | $8 - 5 = 3$ |
| Tea only | $40 - 10 - 5 - 5 = 20$ |
| Coffee only | $35 - 10 - 3 - 5 = 17$ |
| Juice only | $25 - 5 - 3 - 5 = 12$ |
| None | $100 - (20 + 10 + 5 + 17 + 3 + 5 + 12) = 28$ |
🚨 Common Mistakes#
- Not starting from the middle: ALWAYS fill in the intersection first. If you start from the outside, you’ll double-count.
- Confusing $P(A)$ with $P(A \text{ only})$: $P(A)$ includes the overlap! $P(A \text{ only}) = P(A) - P(A \cap B)$.
- Forgetting the “neither” region: The four regions of a two-set Venn diagram must add up to $n(S)$.
- Addition rule without subtracting: $P(A \cup B) \neq P(A) + P(B)$ unless the events are mutually exclusive.
- Not checking totals: After filling in the diagram, all regions must sum to the total sample space.
💡 Pro Tip: The “Algebra” Approach#
If you’re given $P(A \cup B)$, $P(A)$, and $P(B)$, you can find the overlap by rearranging:
$P(A \cap B) = P(A) + P(B) - P(A \cup B)$
This is just the addition rule solved for the intersection.
🔗 Related Grade 10 topics:
- Probability Basics — fundamental probability rules, complement, and relative frequency
- Solving Equations — solving for unknowns in Venn diagrams uses algebra
📌 Where this leads in Grade 11:
- Independent & Dependent Events — tree diagrams, without replacement, and the test for independence