Probability — Grade 11 & 12 Maths (CAPS)

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)

💡

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) = 23/40 = 0,575 ≈ 0,58.
A includes the overlap. 23 ÷ 40 is 0,575 — to two decimals that rounds UP to 0,58.
3
P(A and B) = 8/40 = 0,20.
Only the overlap.
4
P(A or B) = 35/40 = 0,875 ≈ 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) = 23/40, P(B) = 20/40, P(A and B) = 8/40.
n(B) = 8 + 12 = 20.
2
P(A or B) = 23/40 + 20/40 − 8/40 = 35/40.
Subtract the overlap. Stay in fortieths: rounding each probability first gives 0,58 + 0,50 − 0,20 = 0,88 only by luck, and 0,57 + 0,50 − 0,20 = 0,87 loses the mark.
3
= 0,875 ≈ 0,88 ✓
Matches the count from the diagram. Round ONCE, at the end.

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.

6Relative frequency and the theoretical probability

Grade 10 introduced the comparison; Grade 11 puts it to work as evidence. Theoretical probability is calculated from the situation; relative frequency is counted from an experiment.

relative frequency = (number of times the event occurred)/(number of trials)

The two tend to settle close together as the number of trials grows — which is what makes relative frequency useful as a test. A die rolled 600 times that gives a six 180 times has a relative frequency of 0,3 against a theoretical 0,167. Over that many trials, that gap is evidence the die is biased.

7Venn diagrams with three events

The new work in Grade 11 is a third event. Three overlapping circles create eight regions, and the only reliable method is to fill them from the middle outwards.

Worked ExampleIn a group of 60 learners: 30 take Maths, 25 take Science, 22 take Geography; 12 take Maths and Science, 9 take Maths and Geography, 8 take Science and Geography, and 5 take all three. How many take none?

1
Start in the centre: all three = 5.
Always fill the innermost region first — every other region depends on it.
2
Maths and Science only = 12 − 5 = 7
The 12 includes the 5 in the middle.
3
Maths and Geography only = 9 − 5 = 4; Science and Geography only = 8 − 5 = 3
Same subtraction for each pair.
4
Maths only = 30 − 7 − 4 − 5 = 14
Take the three overlapping regions off the total for Maths.
5
Science only = 25 − 7 − 3 − 5 = 10; Geography only = 22 − 4 − 3 − 5 = 10
6
Total in at least one = 14 + 10 + 10 + 7 + 4 + 3 + 5 = 53
Add all seven regions.
7
None = 60 − 53 = 7
The outside region.

⚠️

Writing 12 into the Maths-and-Science overlap. That 12 is the total for Maths and Science, and 5 of them also take Geography — so only 7 belong in the two-circle region. Filling outside-in is what causes this error; always start at the centre.

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Check your diagram by adding every region, including the outside. If it does not equal the total in the question, something has been double-counted.

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).
Show answer ▾Hide answer ▴
15 + 8 = 23
2
Using the Venn diagram below, calculate P(A and B).
Show answer ▾Hide answer ▴
8/40 = 0,20
3
Using the Venn diagram below, calculate P(A or B).
Show answer ▾Hide answer ▴
(15 + 8 + 12)/40 = 35/40 = 0,88
4
Using the Venn diagram below, calculate the probability that a learner takes neither A nor B.
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5/40 = 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.
9
A coin is tossed 400 times and lands heads 214 times. Calculate the relative frequency of heads and comment on whether the coin appears fair.
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214 ÷ 400 = 0,535 ✓; close to the theoretical 0,5 over a large number of trials, so the coin appears fair ✓
10
In a class of 40: 22 play soccer, 18 play netball, 15 play tennis; 9 play soccer and netball, 7 play soccer and tennis, 6 play netball and tennis, and 4 play all three. How many play none of the three?
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Centre 4 ✓; pairs only: 5, 3, 2 ✓; singles: soccer 22 − 5 − 3 − 4 = 10, netball 18 − 5 − 2 − 4 = 7, tennis 15 − 3 − 2 − 4 = 6 ✓; total in at least one = 10 + 7 + 6 + 5 + 3 + 2 + 4 = 37, so none = 40 − 37 = 3 ✓
11
Explain why you must fill a three-event Venn diagram from the CENTRE outwards.
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Each pairwise total already includes the learners who take all three ✓; filling from the centre lets you subtract that overlap so no one is counted twice ✓