Probability — Grade 10 Maths (CAPS)

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

Also known as: Venn diagrams · addition rule · mutually exclusive events

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

1The basics

When every outcome is equally likely: P(E) = (number of favourable outcomes)/(total number of outcomes) · always: 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) = 8/35.
Read the middle region directly.
2
A or B = 12 + 8 + 10 = 30, so P(A or B) = 30/35 = 6/7.
Add all three regions inside the circles.
3
Check with the rule: 20/35 + 18/35 − 8/35 = 30/35 ✓
The addition rule gives the same answer.
4
Neither = 5, so P(neither) = 5/35 = 1/7.
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.

4Relative frequency vs theoretical probability

There are two ways to put a number on a probability, and the ATP asks you to compare them.

When every outcome is equally likely: theoretical P(A) = (number of favourable outcomes)/(total number of outcomes) relative frequency = (number of times A happened)/(number of trials)

Theoretical probability is worked out from the situation itself, before anything is done. A fair die has six equally likely faces, so P(4) = 1/6 — no dice need be rolled.
Relative frequency (also called experimental probability) comes from actually doing it and counting. Roll a die 60 times, get twelve 4s, and the relative frequency is 12/60 = 1/5.

They rarely match exactly, and that is expected. The key idea is that as the number of trials grows large, the relative frequency tends to settle close to the theoretical probability — not closer after every single trial, because chance keeps nudging it about, but closer on the whole. Twelve 4s in 60 rolls is unremarkable; 12 000 in 60 000 would say the die is loaded.

Worked ExampleA drawing pin is dropped 200 times and lands point-up 130 times. Determine the relative frequency, and explain why the probability cannot be found by counting equally likely outcomes.

1
Relative frequency = 130/200 = 0,65
Times it happened over number of trials.
2
The two outcomes are not known to be equally likely — a pin is not symmetrical, so point-up and point-down need not have the same chance.
The favourable-over-total rule only works for equally likely outcomes.
3
So that rule cannot give this probability, and with no other information to go on we estimate it by experiment: P(point-up) ≈ 0,65.
This is exactly when relative frequency is the tool to use.

💡

Use the comparison as a fairness test. If a coin is tossed 500 times and lands heads 251 times, the relative frequency 0,502 sits right on the theoretical 0,5 — good evidence the coin is fair. A relative frequency far from the theoretical value, over many trials, is evidence that something is biased.

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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1/6
2
Find P(rolling an even number) on a fair die.
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3/6 = 1/2
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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8/35
7
From the Venn diagram, find P(A or B).
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30/35 = 6/7
8
From the Venn diagram, find P(neither A nor B).
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5/35 = 1/7
9
n(A) = 20 and 8 are in both. How many take ONLY A?
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20 − 8 = 12
10
A die is rolled 90 times and a 5 comes up 12 times. Calculate the relative frequency of rolling a 5.
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12/90 ✓ = 2/15 ≈ 0,13 ✓
11
For that same die, write down the theoretical probability of rolling a 5 and comment on the difference.
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1/6 ≈ 0,17 ✓; the relative frequency 0,13 is a little lower, which is normal variation over only 90 rolls ✓
12
What generally happens to the relative frequency as the number of trials becomes very large?
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It tends to settle close to the theoretical probability ✓ — though it need not move closer after every single trial, because chance variation remains ✓
13
Give ONE example of a situation where the probability CANNOT be found by counting equally likely outcomes, and say why.
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Dropping a drawing pin or a bottle top ✓ — the outcomes are not equally likely, so there are no equal outcomes to count; the probability is estimated from an experiment instead (relative frequency) ✓