Grade 9 Mathematics
Grade 9 · Term 3Mathematics

Probability

We study theoretical and experimental probability, Venn diagrams, two-way tables, and tree diagrams for compound events. We compare relative frequency (experimental) with theoretical probability and apply addition rules.

8.1 Single Events and Probability Rules8.2 Compound Events — Tables and Tree Diagrams
Week 9

8.1 Single Events and Probability Rules

  • Define probability and the sample space
  • Calculate theoretical probability using P(E) = n(E)/n(S)
  • Apply the complement rule: P(not E) = 1 − P(E)
  • Compare theoretical and experimental (relative frequency) probability
  • Use Venn diagrams to solve probability problems involving overlapping events
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Real-World Connection

Probability governs insurance premiums, weather forecasting, casino games, and medical diagnosis. A weather app says '70% chance of rain' — that's probability based on historical patterns. An insurance company calculates the probability you'll have an accident to set your premium. Every decision under uncertainty uses probability.

Definition

Sample Space (S)

The set of ALL possible outcomes of an experiment.

e.g. Rolling a die: S={1,2,3,4,5,6},  n(S)=6\text{e.g. Rolling a die: } S = \{1, 2, 3, 4, 5, 6\}, \; n(S) = 6

Theoretical Probability

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

n(E) = number of favourable outcomes; n(S) = total outcomes (all equally likely)

Complement Rule

P(not E)=1−P(E)P(\text{not } E) = 1 - P(E)

The probability of an event NOT happening = 1 minus the probability it does happen

Property / Rule

Probability Scale

Probability is always between 0 and 1 inclusive. P = 0 means impossible; P = 1 means certain.

0≤P(E)≤10 \leq P(E) \leq 1

Definition

Relative Frequency

Experimental probability: frequency of an event divided by the number of trials. Approaches theoretical probability as trials increase.

Relative frequency=number of times event occurredtotal number of trials\text{Relative frequency} = \frac{\text{number of times event occurred}}{\text{total number of trials}}

Definition

Venn Diagram

A diagram using overlapping circles inside a rectangle to display events and their relationships. The rectangle represents the sample space S. The overlap (intersection) shows outcomes belonging to BOTH events. Regions outside all circles show outcomes in neither event.

A∩B=both eventsA∪B=at least one eventA′=not A (complement)A \cap B = \text{both events} \qquad A \cup B = \text{at least one event} \qquad A' = \text{not A (complement)}

Addition Rule for Probability

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

Subtract P(A∩B) once to avoid double-counting the overlap. If A and B are mutually exclusive, P(A∩B)=0 and the rule simplifies to P(A)+P(B).

Worked Examples

Worked Example

Calculating probability

Problem

A bag contains 4 red, 3 blue, and 5 green marbles. A marble is drawn at random. Find: (a) P(red) (b) P(not blue) (c) P(red or green)

Worked Example

Comparing theoretical and experimental probability

Problem

A die is rolled 60 times. It lands on 4 exactly 12 times. (a) Find the experimental probability of rolling a 4. (b) Compare with the theoretical probability. (c) How would the result change with 600 rolls?

Worked Example

Listing the sample space

Problem

A coin is flipped and a die is rolled. List the sample space and find P(heads and even number).
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 fair die is rolled. Find P(even number).
2
L1 · Knowledge1 mark
The probability of rain tomorrow is 0.35. What is the probability it will NOT rain?
3
L2 · Routine Procedures4 marks
A card is drawn from a standard 52-card deck. Find: (a) P(heart) (b) P(face card) (c) P(ace).
4
L2 · Routine Procedures3 marks
A coin is flipped 100 times and lands on heads 43 times. Find the relative frequency of heads. How does this compare to theoretical probability?
5
L3 · Complex Procedures5 marks
In a class of 30, 18 play soccer, 12 play netball, and 6 play both. Draw a Venn diagram and find P(plays at least one sport).
6
L3 · Complex Procedures4 marks
A bag has 5 red and 3 blue balls. After 200 trials (with replacement), red appeared 128 times. Is the bag's selection biased? Explain using both theoretical and experimental probability.
7
L4 · Problem Solving4 marks
P(A) = 0.4, P(B) = 0.3, P(A and B) = 0.1. Find P(A or B) and P(neither A nor B).
8
L4 · Problem Solving5 marks
Two events are mutually exclusive with P(A)=0.3 and P(B)=0.45. Find P(A or B), P(A and B), and P(not A and not B).
Week 10

8.2 Compound Events — Tables and Tree Diagrams

  • Use two-way tables to determine probabilities of compound events
  • Draw tree diagrams to list all possible outcomes
  • Calculate probabilities of compound events using tree diagrams
🌍

Real-World Connection

Tree diagrams are used in decision analysis and genetics. A geneticist uses a Punnett square (a two-way table) to predict the probability that a child inherits a trait. A project manager uses a decision tree to map out possible outcomes and their probabilities when making business decisions. A sports analyst uses them to calculate match outcome probabilities.

Property / Rule

Probability Along a Path

For independent events, multiply probabilities along a branch of a tree diagram.

P(A then B)=P(A)×P(B)(independent events)P(A \text{ then } B) = P(A) \times P(B) \quad (\text{independent events})

ℹ️ Note

The probabilities of ALL branches from any point must add up to 1. If one branch has probability 0.3 and another has 0.5, there must be a third branch with probability 0.2.

Worked Examples

Worked Example

Tree diagram for two coin flips

Problem

A fair coin is flipped twice. Draw a tree diagram and find: (a) P(two heads) (b) P(at least one tail).

Worked Example

Two-way table for compound events

Problem

A spinner has sections labelled 1, 2, 3. A die is rolled. Draw a two-way table showing all outcomes. Find P(sum > 6) and P(spinner shows 2 and die shows even).

Worked Example

Tree diagram with unequal probabilities

Problem

A bag has 3 red and 2 blue balls. One ball is drawn, its colour noted, then replaced. A second draw is made. Find P(same colour both times).
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
List all outcomes when a coin and a die are tossed together.
2
L1 · Knowledge1 mark
Two dice are rolled. How many elements are in the sample space?
3
L2 · Routine Procedures4 marks
A bag has 3 red and 2 blue balls. A ball is drawn, replaced, then drawn again. Find P(both same colour).
4
L2 · Routine Procedures4 marks
A two-way table shows: 15 boys play sport, 8 boys don't; 12 girls play sport, 10 girls don't. Of the learners who play sport, what fraction are girls? Show your working using the two-way table.
5
L3 · Complex Procedures5 marks
Bag A has 4 red, 1 blue. Bag B has 2 red, 3 blue. A bag is chosen at random, then one ball drawn. Find P(red ball).
6
L3 · Complex Procedures5 marks
Three coins are flipped. Using a tree diagram, find P(exactly 2 heads).
7
L4 · Problem Solving5 marks
A school raffle has 50 tickets numbered 1–50. Find the probability that the winning ticket is a multiple of 3 OR greater than 40.
8
L4 · Problem Solving5 marks
In a town, 60% of households have a dog and 40% have a cat. 30% have both. A household is selected at random. Find the probability that it has a dog or a cat but NOT both (exclusive or).

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Data Handling

Probability Grade 9 Maths CAPS Notes & Examples | MathSciBuddy