Tree Diagrams for Multi-Step Experiments

Tree diagrams come alive when students physically manipulate outcomes. Constructing diagrams with hands-on materials helps Year 10 students visualize how probabilities shift across dependent events, building intuition before abstract calculation. Active construction reduces common errors in path multiplication and replacement scenarios.

• Year 10
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Construct tree diagrams to systematically list all possible outcomes for multi-step probability experiments.
2. Calculate the probability of compound events occurring in sequence, with and without replacement.
3. Analyze how the removal of an item affects subsequent probabilities in dependent events.
4. Justify the multiplication of probabilities along branches of a tree diagram based on event independence or dependence.

Escape Room

⏱ 35 min·Pairs

Pairs Build: Marble Bag Draws

Provide bags with colored marbles. Pairs draw twice without replacement, sketch tree diagrams on mini-whiteboards, label branches with fractions, and calculate probabilities for specific color sequences. Pairs share one diagram with the class for peer feedback.

Prepare & details

Explain how the sample space changes when an item is not replaced after the first draw.

Facilitation Tip: During the Pairs Build activity, circulate and ask each pair to explain one branch’s probability change after their first draw to uncover misconceptions immediately.

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: Dice Chain Challenge

Groups roll two dice sequentially, with replacement on even rolls only. They construct tree diagrams showing conditional branches, compute paths to sums greater than 10, and test 20 trials to verify predictions. Compare group diagrams on chart paper.

Prepare & details

Justify why we multiply probabilities along the branches of a tree diagram.

Facilitation Tip: In the Small Groups Dice Chain Challenge, require groups to present their tree and probability calculations to the class before moving to the next scenario.

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

⏱ 30 min·Whole Class

Whole Class: Spinner Relay

Divide spinners into sections with varying probabilities. Students relay to spin twice without replacement, updating a class tree diagram on the board after each turn. Calculate cumulative probabilities live and discuss adjustments.

Prepare & details

Construct a tree diagram for a multi-step experiment involving different outcomes at each stage.

Facilitation Tip: For the Whole Class Spinner Relay, have each student physically spin the spinner once to collect class data, then compare individual results to the theoretical tree 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: Card Sort Trees

Give students shuffled cards representing outcomes. They sort into tree diagram templates for draws with and without replacement, label probabilities, and solve for event chances. Submit for quick teacher check.

Prepare & details

Explain how the sample space changes when an item is not replaced after the first draw.

Facilitation Tip: During the Card Sort Trees activity, check that students group dependent and independent events correctly before calculating 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

Teaching This Topic

Start with concrete manipulatives to build foundational understanding before moving to abstract diagrams. Avoid rushing to formulas; let students discover why probabilities multiply along paths through repeated trials. Research shows that students who construct their own diagrams retain concepts longer and apply them to new contexts more successfully. Prioritize peer explanation over teacher correction to deepen understanding.

What to Expect

By the end of these activities, students should accurately map multi-step experiments, calculate dependent probabilities, and explain why probabilities change without replacement. Diagrams will show correct branching and labeled probabilities. Students will justify their calculations using the tree structure.

Watch Out for These Misconceptions

Common Misconception

During Pairs Build activity, watch for students who keep the same marble counts on all branches, ignoring the reduction after the first draw.

What to Teach Instead

Prompt pairs to recount the marbles aloud after each draw and adjust the counts on their tree diagram before calculating probabilities.

Common Misconception

During Small Groups Dice Chain Challenge, watch for students who multiply all probabilities by the number of steps instead of along paths.

What to Teach Instead

Have groups trace a single outcome path with their fingers while explaining each multiplication step to the class.

Common Misconception

During Card Sort Trees activity, watch for students who treat dependent and independent events the same way, using fixed probabilities for both.

What to Teach Instead

Ask students to sort the cards into two piles (dependent vs. independent) before calculating, and justify their sorting to a partner.

Assessment Ideas

Quick Check

After the Pairs Build activity, provide each pair with a marble scenario (e.g., 4 green and 1 yellow marble, two draws without replacement) and have them construct a tree diagram and calculate the probability of drawing two greens. Collect one diagram per pair to assess accuracy in branching and probability calculation.

Discussion Prompt

During the Whole Class Spinner Relay, pause after collecting class data and ask, 'Explain why the probability of landing on red changed from the first to the second spin, using the tree diagram we built.' Facilitate a class discussion where students use the diagram to justify dependent events.

Exit Ticket

After the Card Sort Trees activity, ask students to write a brief explanation justifying why probabilities are multiplied along the branches of a tree diagram for compound events. They should include a simple example from their sorted cards to illustrate their reasoning.

Extensions & Scaffolding

• Challenge students to create a tree diagram for a three-step experiment (e.g., drawing three marbles without replacement) and calculate all possible probabilities.
• Scaffolding: Provide pre-labeled diagrams with missing probabilities for students to complete during the Card Sort Trees activity.
• Deeper: Ask students to design their own multi-step experiment, collect class data, and compare empirical results to theoretical tree probabilities in a follow-up lesson.

Key Vocabulary

Sample Space	The set of all possible outcomes of a probability experiment. For multi-step experiments, this includes all combinations of outcomes from each stage.
Tree Diagram	A visual tool used to represent the outcomes of a probability experiment. It consists of branches showing possible events and their probabilities at each stage.
Compound Event	An event that consists of two or more simple events. In tree diagrams, this is represented by a path from the start to an end point.
Dependent Events	Events where the outcome of one event affects the probability of the other event. This occurs in experiments without replacement.
Independent Events	Events where the outcome of one event does not affect the probability of the other event. This occurs in experiments with replacement.