Why is "Adding the outcomes of two events instead of multiplying them." a common mistake in Probability?

Why it happens: Students think two dice with 6 faces each give 12 outcomes. How to avoid it: Multiply the outcomes of each event. Two dice give 6 × 6 = 36 combined outcomes.

Why is "Adding the probabilities along a single branch of a tree diagram." a common mistake in Probability?

Why it happens: It is easy to confuse the two tree-diagram rules. How to avoid it: Multiply *along* the branches of one path; only add the probabilities of *separate* paths.

Why is "Forgetting that a result can happen by more than one path." a common mistake in Probability?

Why it happens: Students find one path and stop, missing the others. How to avoid it: Check every path. 'Exactly one red' includes red-then-blue *and* blue-then-red.

Why is "Listing combined outcomes randomly and missing or repeating some." a common mistake in Probability?

Why it happens: Without a system, it is hard to track which outcomes are covered. How to avoid it: List systematically — work fully through one event before moving to the next.

Why is "Deciding a dice is biased after only a few trials." a common mistake in Probability?

Why it happens: A small number of trials can look very uneven by chance. How to avoid it: Experimental probability only settles near the theoretical value after many trials.

Systematic listing for a coin and a dice

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Probability — Cambridge Lower Secondary Maths, Grade 8

You already know theoretical and experimental probability and sample space diagrams — this guide goes further. You will work confidently with combined events, build clear sample space diagrams, and read tree diagrams to find the probability of a sequence of outcomes.

At a glance

Every probability is a number from 0 (impossible) to 1 (certain).
Theoretical probability comes from reasoning; experimental probability comes from trials.
A combined event is made of two or more separate events happening.
A sample space diagram lists every possible combined outcome in a grid.
A tree diagram shows a sequence of events as branching paths.
Multiply the probabilities along a branch to find the chance of that path.
Add the probabilities of the separate paths that give the result you want.
The probabilities on the branches from any single point always add up to 1.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 9 — Find the probability of combined events using systematic listing and sample space diagrams.
Stage 9 — Construct and interpret tree diagrams for sequences of two events.
Stage 9 — Multiply probabilities along the branches of a tree diagram and add path probabilities.
Stage 9 — Compare theoretical and experimental probability for combined events.

A quick recap of probability

Probability is a number from 0 to 1, found by reasoning or by trials.

Stage 9 probability builds on what you already know, so a quick refresher first.

Every probability is a number on a scale from 0 to 1. A probability of 0 means impossible, 1 means certain, and $\tfrac{1}{2}$ means an even chance.

For equally likely outcomes: $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$

Combined events

A combined event is two or more separate events happening together.

A combined event happens when two or more events occur together — flipping a coin and rolling a dice, or picking a counter and then picking another.

The safest way to handle a combined event is to list every possible outcome in an organised way, so none is missed and none is repeated. This is called systematic listing.

A neat list of all 12 combined outcomes — none missed, none repeated.

For a coin and a dice, list the coin's results in order, then pair each with the dice's $1$ to $6$ . That gives outcomes.

Sample space diagrams for combined events

A sample space diagram is a grid of every combined outcome.

A sample space diagram is a clear grid that lists every combined outcome of two events — one event along the top, the other down the side, with each cell showing one outcome.

Tree diagrams

A tree diagram shows a sequence of events as branching paths.

A tree diagram shows a sequence of events as a set of branching paths. Each event adds a new set of branches, and every branch is labelled with its probability.

Each path is one outcome; multiply along a branch to find its probability.

There are two golden rules for tree diagrams:

Multiply along the branches. The probability of a whole path is found by multiplying the probabilities on each branch of that path. For Red then Red: $0.3 \times 0.3 = 0.09$ .

Comparing theoretical and experimental probability

Experimental results drift towards theoretical probability over many trials.

For combined events, you can find probability two ways — by reasoning (theoretical) and by doing trials (experimental) — and it is worth comparing them.

Theoretical: rolling two dice, a sample space diagram shows $P(\text{total} = 7) = \tfrac{6}{36} = \tfrac{1}{6} \approx 0.167$ .

Where you'll use this next

Combined events and tree diagrams lead into all later probability work.

These ideas set you up well for the probability ahead — and for thinking clearly about chance:

Later probability work — you will go on to handle longer sequences and events where one outcome changes the next.
Fractions, decimals and percentages — every probability is written in one of these forms, so this topic gives them a real purpose.
Data handling — experimental probability comes straight from trials recorded in frequency tables.
In everyday life, combined-event reasoning shapes games, weather forecasts, insurance and sensible decision-making.

The two golden rules — multiply along branches, add the paths — are the heart of all the probability you will meet later. Practise them on tree diagrams now, and harder probability will feel familiar when it arrives.

Later probability extends to longer sequences of events.
Probabilities are written as fractions, decimals or percentages.
Experimental probability comes from recorded trials.
Multiply along branches and add the paths — the rules carry forward.

Quick recap

Probability runs from 0 (impossible) to 1 (certain).
A combined event is two or more events happening together.
Total combined outcomes = multiply the outcomes of each event.
A sample space diagram lists every combined outcome in a grid.
A tree diagram shows a sequence of events as branching paths.
Multiply along the branches; add the probabilities of separate paths.
More trials bring experimental probability closer to theoretical.

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 9). Last reviewed 2026-05-19.

Step-by-step worked examples — Probability

Step-by-step solutions for probability, written exactly the way a tutor would explain them at the board.

1Counting combined outcomes
Getting started• combined event, counting
Question
A spinner has $4$ equal sections and a coin has $2$ sides. How many combined outcomes are there when you spin the spinner and flip the coin?
Step-by-step solution
1. Step 1
  The total number of combined outcomes is found by multiplying the outcomes of each event.
2. Step 2
  The spinner has $4$ outcomes and the coin has $2$ outcomes.
  $\text{total} = 4 \times 2$
3. Step 3
  Work out the multiplication.
  $\text{total} = 8$
Answer
There are $8$ combined outcomes.
2Using a sample space diagram
Getting started• sample space diagram, two dice
Question
Two ordinary dice are rolled and their scores added. Using a sample space diagram, find the probability that the total is $5$ .
Step-by-step solution
1. Step 1
  A sample space diagram for two dice has $6 \times 6 = 36$ equally likely outcomes.
3Multiplying along a tree diagram
Building confidence• tree diagram, combined event
Question
A bag has counters, and the probability of picking a green counter is $0.4$ . A counter is picked, replaced, then a second is picked. Find the probability that both counters are green.
Step-by-step solution
1. Step 1
  Each pick has probability $0.4$ for green. Draw the branch for green then green.
4Adding paths on a tree diagram
Building confidence• tree diagram, adding paths
Question
A spinner lands on red with probability $0.3$ and blue with probability $0.7$ . It is spun twice. Find the probability of getting exactly one red.
Step-by-step solution
1. Step 1
  Exactly one red can happen two ways: red then blue, or blue then red.
2. Step 2
5Finding 'at least one' the quick way
Stretch• tree diagram, at least one
Question
A coin is biased so that $P(\text{heads}) = 0.6$ . It is flipped twice. Find the probability of getting at least one head.
Step-by-step solution
1. Step 1
  'At least one head' is the opposite of 'no heads at all', which means tails then tails.

Probability — Cambridge Lower Secondary Mathematics — Stage 9 Revision Notes & Practice | Tutopiya

Probability

Detailed Study Notes

Take this whole topic with you

Probability — Cambridge Lower Secondary Maths, Grade 8

At a glance

What you’ll learn

A quick recap of probability

Combined events

Sample space diagrams for combined events

Tree diagrams

Comparing theoretical and experimental probability

Where you'll use this next

Quick recap

Step-by-step worked examples — Probability

1Counting combined outcomes

2Using a sample space diagram

3Multiplying along a tree diagram

4Adding paths on a tree diagram

5Finding 'at least one' the quick way

Key Definitions and Keywords — Probability

Probability

Combined event

Sample space diagram

Tree diagram

Theoretical probability

Experimental probability

Outcome

Trial

Systematic listing

Equally likely

Biased

Event

Common Mistakes and Misconceptions — Probability

✕Adding the outcomes of two events instead of multiplying them.

✕Adding the probabilities along a single branch of a tree diagram.

✕Forgetting that a result can happen by more than one path.

✕Listing combined outcomes randomly and missing or repeating some.

✕Deciding a dice is biased after only a few trials.