Probability: Venn Diagrams & Rules (Grade 11)

The addition rule, mutually exclusive and complementary events, reading and filling in Venn diagrams, and testing whether two events are independent.

1The rules you must know

P(A or B) = P(A) + P(B) − P(A and B)

Mutually exclusive: P(A and B) = 0
Complementary: P(A) + P(not A) = 1
Independent: P(A and B) = P(A) × P(B)

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You subtract P(A and B) in the addition rule because the overlap gets counted twice — once in P(A) and once in P(B).

2Reading a Venn diagram

40 learners. 15 take only A, 8 take both, 12 take only B, and 5 take neither.

Worked ExampleFrom the diagram, calculate P(A), P(A and B) and P(A or B)

1
Total = 15 + 8 + 12 + 5 = 40.
Add every region — the ones outside the circles too.
2
n(A) = 15 + 8 = 23, so P(A) = 2340 = 0,57.
A includes the overlap.
3
P(A and B) = 840 = 0,20.
Only the overlap.
4
P(A or B) = 3540 = 0,88.
15 + 8 + 12 = 35, everything inside the circles.

⚠️

The number written inside the overlap is only the 'both' group. n(A) is the overlap plus the A-only region — here 23, not 15.

3Checking the addition rule

Worked ExampleVerify P(A or B) using the rule

1
P(A) = 0,57, P(B) = 2040 = 0,50.
n(B) = 8 + 12 = 20.
2
P(A or B) = 0,57 + 0,50 − 0,20.
Subtract the overlap.
3
= 0,88 ✓
Matches the count from the diagram.

4Mutually exclusive vs complementary

Mutually exclusive — they cannot both happen. The circles do not overlap. P(A and B) = 0.
Complementary — mutually exclusive and they cover everything. P(A) + P(B) = 1.

Every pair of complementary events is mutually exclusive, but not the other way round: rolling a 1 and rolling a 2 are mutually exclusive, yet they do not add to 1.

5Testing independence

Worked ExampleP(A) = 0,4 P(B) = 0,5 P(A and B) = 0,2. Are A and B independent?

1
P(A) × P(B) = 0,4 × 0,5 = 0,2.
Work out the product first.
2
P(A and B) = 0,2.
Compare with the given value.
3
They are equal, so yes, A and B are independent.
State the conclusion in words — that is where the mark is.

⚠️

'Independent' and 'mutually exclusive' are not the same thing. In fact if two events with non-zero probability are mutually exclusive, they cannot be independent.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
In the Venn diagram below, 40 learners were surveyed. Determine n(A).
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15 + 8 = 23
2
Using the Venn diagram below, calculate P(A and B).
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840 = 0,20
3
Using the Venn diagram below, calculate P(A or B).
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15 + 8 + 1240 = 3540 = 0,88
4
Using the Venn diagram below, calculate the probability that a learner takes neither A nor B.
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540 = 0,13
5
P(A) = 0,4, P(B) = 0,5 and P(A and B) = 0,2. Are A and B independent? Show your working.
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P(A)×P(B) = 0,4×0,5 = 0,2 = P(A and B), so yes, they are independent.
6
P(A) = 0,3, P(B) = 0,45 and A and B are mutually exclusive. Calculate P(A or B).
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0,3 + 0,45 − 0 = 0,75
7
If P(A) = 0,62, calculate P(not A).
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1 − 0,62 = 0,38
8
Explain the difference between mutually exclusive and complementary events.
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Mutually exclusive events cannot happen together, so P(A and B) = 0. Complementary events are mutually exclusive AND cover the whole sample space, so P(A) + P(B) = 1.