Probability: Venn Diagrams & the Addition Rule (Grade 10)

Sample spaces, complementary and mutually exclusive events, the addition rule, and reading a two-set Venn diagram to answer probability questions.

Grade 10 probability moves from single events to combinations of events — and a Venn diagram is nearly always the fastest way in.

1The basics

P(E) = number of favourable outcomestotal number of outcomes · 0 ≤ P(E) ≤ 1

The sample space (S) is every possible outcome.
Complementary events: P(not A) = 1 − P(A). They must add to 1.
Mutually exclusive events cannot happen together, so P(A and B) = 0.

2The addition rule

P(A or B) = P(A) + P(B) − P(A and B)

You subtract the overlap because anything in both sets would otherwise be counted twice. If the events are mutually exclusive the overlap is 0, so the rule shortens to P(A) + P(B).

3Reading a Venn diagram

35 learners. 20 take A, 18 take B, 8 take both, and 5 take neither.

Worked ExampleFrom the diagram, find P(A and B), P(A or B) and P(neither)

1
The overlap holds 8 learners, so P(A and B) = 835.
Read the middle region directly.
2
A or B = 12 + 8 + 10 = 30, so P(A or B) = 3035 = 67.
Add all three regions inside the circles.
3
Check with the rule: 2035 + 1835 − 835 = 3035 ✓
The addition rule gives the same answer.
4
Neither = 5, so P(neither) = 535 = 17.
That is the region outside both circles.

⚠️

When a question says 20 learners take A, that includes the ones who also take B. The A-only region is 20 − 8 = 12 — fill the overlap in first, then work outwards.

💡

Always check that every region adds up to n(S). Here 12 + 8 + 10 + 5 = 35 ✓ — if it does not, a region is wrong.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
A fair die is rolled. Find P(rolling a 4).
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16
2
Find P(rolling an even number) on a fair die.
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36 = 12
3
If P(A) = 0,3 find P(not A).
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1 − 0,3 = 0,7
4
State the addition rule.
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P(A or B) = P(A) + P(B) − P(A and B)
5
If A and B are mutually exclusive, what is P(A and B)?
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0
6
From the Venn diagram, find P(A and B).
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835
7
From the Venn diagram, find P(A or B).
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3035 = 67
8
From the Venn diagram, find P(neither A nor B).
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535 = 17
9
n(A) = 20 and 8 are in both. How many take ONLY A?
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20 − 8 = 12