Probability of Independent Events using Tree Diagrams

Tree diagrams make abstract independent events concrete for Year 8 students by turning multiplication of probabilities into visual paths and labeled branches. Active construction cements the link between sample spaces that grow multiplicatively and the multiplication rule for independent events.

• Year 8
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Construct a tree diagram to accurately represent the outcomes of two independent events.
2. Calculate the probability of compound events by multiplying probabilities along the paths of a tree diagram.
3. Explain how the sample space expands multiplicatively when a second independent event is added to an experiment.
4. Analyze the structure of a tree diagram to identify all possible combined outcomes for a two-stage experiment.

Ready-to-Use Activities

Escape Room

⏱ 30 min·Pairs

Pairs: Dice Roll Trees

Pairs roll two dice 24 times and record outcomes on a table. They then construct a tree diagram for all possibilities, label branch probabilities, and calculate chances like double six. Compare experimental frequencies to theoretical values and adjust trees if needed.

Prepare & details

Explain how the sample space changes when we add a second stage to an experiment.

Facilitation Tip: During Pairs: Dice Roll Trees, circulate and ask each pair to explain why the probability on the second branch stays the same as the first for an independent roll.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 45 min·Small Groups

Small Groups: Spinner Experiments

Groups design two spinners with four sectors each, spin twice, and build a tree diagram. Calculate probabilities for specific colour combinations. Run 30 trials to test predictions and graph results for class sharing.

Prepare & details

Analyze how tree diagrams can help us organize complex decision-making processes.

Facilitation Tip: While Small Groups: Spinner Experiments build, place a checklist on each table to ensure students include every possible outcome, even the less likely ones.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 40 min·Whole Class

Whole Class: Weather Prediction Trees

Project a large tree diagram on the board for two-day rain probabilities. Class votes on branch outcomes, multiplies paths, and simulates with random draws. Discuss how trees organise multi-step forecasts.

Prepare & details

Construct a tree diagram to represent the outcomes of two independent events.

Facilitation Tip: In Whole Class: Weather Prediction Trees, assign each table a different starting probability so groups see how varied initial values change final compound probabilities.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Escape Room

⏱ 25 min·Individual

Individual: Digital Tree Builder

Students use online tools to create trees for coin flips and card draws. Input probabilities, generate sample spaces, and solve for event chances. Share screenshots in a class gallery for peer feedback.

Prepare & details

Explain how the sample space changes when we add a second stage to an experiment.

Facilitation Tip: For Individual: Digital Tree Builder, have students print their final diagrams and exchange with a partner for peer verification of branch labels and calculations.

Setup: Group tables with puzzle envelopes, optional locked boxes

Materials: Puzzle packets (4-6 per group), Lock boxes or code sheets, Timer (projected), Hint cards

• Remember
• Apply
• Analyze
• Relationship Skills
• Self-Management

Teaching This Topic

Start by modeling one simple two-stage tree with dice, emphasizing that each branch represents a mutually exclusive outcome and that the probability on each new branch is independent of the previous one. Avoid rushing to the multiplication rule; instead, have students count outcomes first to see why m × n paths appear. Research shows that building the tree by hand before moving to digital tools strengthens understanding of sample space expansion.

What to Expect

Students will confidently draw accurate tree diagrams, label each branch with correct probabilities, and compute compound event probabilities by multiplying along paths. They will also articulate why the sample space size grows by multiplication across stages.

Watch Out for These Misconceptions

Common Misconception

During Pairs: Dice Roll Trees, watch for students who add branch probabilities instead of multiplying along paths.

What to Teach Instead

Have pairs recount the outcomes along one path and calculate the empirical frequency during repeated trials to see that the product matches the observed frequency, not the sum.

Common Misconception

During Small Groups: Spinner Experiments, watch for incomplete trees that omit less likely outcomes.

What to Teach Instead

Use the provided checklists to verify every sector is represented and ask groups to swap diagrams to spot missing branches before proceeding.

Common Misconception

During Whole Class: Weather Prediction Trees, watch for students who add the number of outcomes at each stage instead of multiplying.

What to Teach Instead

Ask groups to list all ordered pairs of outcomes on the board and count them to confirm m × n total paths, reinforcing the expansion visually.

Assessment Ideas

Quick Check

After Pairs: Dice Roll Trees, display a prompt asking for the probability of rolling a 3 followed by an even number. Collect diagrams to check correct branching, labeling, and multiplication along the path.

Exit Ticket

During Individual: Digital Tree Builder, ask students to submit their diagram and calculation for the probability of a specific two-stage event, such as landing on red twice in a row with replacement.

Discussion Prompt

After Whole Class: Weather Prediction Trees, pose the prompt: ‘How does doubling the number of days in the weather model change the total number of outcome paths?’ Have students discuss in pairs, referencing their tree diagrams to justify the multiplicative growth.

Extensions & Scaffolding

• Challenge students to design a spinner with unequal sectors so that the tree diagram yields three distinct compound probabilities.
• Scaffolding: Provide partially completed tree templates with missing branch labels or probabilities for students to fill in before calculating.
• Deeper exploration: Ask students to compare experimental frequencies from Small Groups: Spinner Experiments with their theoretical probabilities and explain any discrepancies.

Key Vocabulary

Independent Events	Two events are independent if the outcome of the first event does not affect the outcome of the second event.
Tree Diagram	A visual tool used to display all possible outcomes of a sequence of events, with branches representing each possible outcome at each stage.
Sample Space	The set of all possible outcomes of an experiment.
Compound Event	An event that consists of two or more independent events occurring in sequence.
Probability	A measure of how likely an event is to occur, expressed as a number between 0 and 1.