Grade 10 Mathematics
Grade 10 · Term 3Mathematics

Probability

We define theoretical and experimental probability, use Venn diagrams to represent events in a sample space, and apply the addition rule, mutual exclusivity, and the complement rule to solve problems.

Week 8

9.1 Probability — Venn Diagrams & Rules

  • Use probability models; compare relative frequency with theoretical probability
  • Apply the addition rule: $P(A\cup B) = P(A) + P(B) - P(A\cap B)$
  • Use Venn diagrams to solve probability problems
  • Identify mutually exclusive events: $P(A\cap B)=0$; complementary events: $P(A)+P(A')=1$
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Real-World Connection

Insurance companies calculate the probability that a car will be involved in an accident in a given year. They use massive historical data sets (experimental probability) and mathematical models to price your premium. When two risks overlap — like being in an accident AND filing a claim — they apply exactly the addition rule you learn here.

Definition

Theoretical Probability

The probability of an event is a number between 0 and 1 measuring how likely it is to occur. P(E) = 0 means impossible; P(E) = 1 means certain. Theoretical probability assumes equally likely outcomes.

P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}

Definition

Relative Frequency (Experimental Probability)

The relative frequency of an event is the proportion of trials in which it actually occurred. As the number of trials increases, the relative frequency gets closer to the theoretical probability.

Rel. freq.=number of times event occurredtotal number of trials\text{Rel. freq.} = \frac{\text{number of times event occurred}}{\text{total number of trials}}

Addition Rule

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

$P(A\cup B)$ = P(A or B); $P(A\cap B)$ = P(A and B) = overlap

Property / Rule

Mutually Exclusive Events

Events A and B are mutually exclusive if they cannot occur simultaneously: A∩B = ∅, so P(A∩B) = 0. The addition rule simplifies to P(A∪B) = P(A) + P(B).

A∩B=∅⇒P(A∪B)=P(A)+P(B)A\cap B=\emptyset\Rightarrow P(A\cup B)=P(A)+P(B)

Property / Rule

Complementary Events

The complement A' is the event that A does NOT occur. A and A' are mutually exclusive and exhaustive.

P(A)+P(A′)=1⇒P(A′)=1−P(A)P(A)+P(A')=1\Rightarrow P(A')=1-P(A)

ℹ️ Note

Always start a Venn diagram problem by finding the intersection (overlap). Label: intersection only-A region, only-B region, neither region. Then read off any required probability.

Worked Examples

Worked Example

Relative frequency vs theoretical probability

Problem

A fair die is rolled 60 times. The number 4 comes up 12 times. Compare the relative frequency with the theoretical probability. What happens as we increase the number of trials?

Worked Example

Venn diagram with addition rule

Problem

In a class of 40: 25 play soccer (S), 18 play tennis (T), 8 play both. Find P(S∪T)P(S\cup T) and P(neither)P(\text{neither}).

Worked Example

Mutually exclusive events

Problem

A card is drawn from a standard deck. AA = drawing a king, BB = drawing a queen. Are A and B mutually exclusive? Find P(A∪B)P(A\cup B).

Worked Example

Find unknown probability from Venn diagram

Problem

P(A) = 0.6, P(B) = 0.5, P(A∪B) = 0.8. Find P(A∩B) and P(A′∩B′)P(A'\cap B').
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
A coin is tossed 20 times and heads appear 9 times. Write down the relative frequency of heads and the theoretical probability of heads. Are they equal?
2
L1 · Knowledge1 mark
If P(A)=0.45P(A) = 0.45, find P(A′)P(A').
3
L2 · Routine Procedures3 marks
P(A)=0.5P(A)=0.5, P(B)=0.3P(B)=0.3, P(A∩B)=0.1P(A\cap B)=0.1. Find P(A∪B)P(A\cup B).
4
L2 · Routine Procedures2 marks
Events AA and BB are mutually exclusive. P(A)=0.4P(A)=0.4, P(B)=0.35P(B)=0.35. Find P(A∪B)P(A\cup B).
5
L3 · Complex Procedures4 marks
In a survey of 50: 30 like coffee (C), 20 like tea (T), 10 like both. Find P(C only)P(\text{C only}) and P(neither)P(\text{neither}).
6
L3 · Complex Procedures4 marks
A card is drawn from a standard 52-card deck. Find P(red or ace)P(\text{red or ace}).
7
L4 · Problem Solving5 marks
A bag has 4 white and 3 black marbles. Two are drawn WITH replacement. Find P(different colours)P(\text{different colours}).
8
L4 · Problem Solving5 marks
Two dice are rolled. Show the sample space has 36 outcomes and find P(sum=7)P(\text{sum}=7) and P(sum>10)P(\text{sum}>10).

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