Mathematics · Year 11

Active learning ideas

Tree Diagrams for Conditional Probability

Tree diagrams make conditional probability concrete by turning abstract events into visible paths. When students draw and label branches, they transform a confusing rule into a series of manageable steps they can check as they go. Active, hands-on work with real objects and peer talk helps them trust that each label and calculation follows logically from the last.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Probability

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Pairs: Diagnostic Test Trees

Pairs receive probabilities for a medical test's true/false positives and negatives. They construct a two-stage tree diagram, calculate the probability of disease given a positive result, then swap papers to verify each other's work and discuss adjustments. Extend by varying test accuracy rates.

Analyze how tree diagrams visually represent conditional probabilities.

Facilitation Tip

During Pairs: Diagnostic Test Trees, circulate and listen for the student who explains why the second set of branches must change after the first event.

What to look for

Present students with a scenario, such as drawing two marbles from a bag without replacement. Ask them to draw the first two levels of the tree diagram, labeling each branch with the correct probability. Check for accurate initial probabilities and conditional probabilities on the second level.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 45 min · Small Groups

Small Groups: Bead Bag Simulations

Groups get bags with colored beads and build tree diagrams for two draws without replacement. They calculate all path probabilities, perform 20 actual draws to compare empirical results, and adjust trees if needed. Record findings on shared posters.

Predict the total number of outcomes from a multi-stage event using a tree diagram.

Facilitation Tip

In Small Groups: Bead Bag Simulations, ask groups to pause after each draw and update their tree before the next draw to reinforce the conditional update.

What to look for

Provide students with a completed tree diagram for a scenario like a biased coin toss followed by a dice roll. Ask: 'Explain why the probabilities on the second set of branches are different from the first set. How does this diagram help us understand the relationship between the two events?'

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 25 min · Whole Class

Whole Class: Chain Event Build

Project a multi-stage scenario like successive weather events on the board. Class votes on branch probabilities, teacher draws the tree live, and students calculate totals in notebooks. Discuss and correct as a group.

Justify the multiplication rule for probabilities along branches of a tree diagram.

Facilitation Tip

During Whole Class: Chain Event Build, deliberately let one path grow longer than the others so students see that all branch sums at a node must equal 1 regardless of path length.

What to look for

Give students a problem involving two dependent events (e.g., selecting students for a committee without replacement). Ask them to calculate the probability of a specific outcome (e.g., selecting two boys) using their tree diagram and write one sentence explaining their calculation.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 30 min · Individual

Individual: Spinner Dependency Challenges

Students use online spinners or paper models for dependent spins, draw trees for three events, compute required probabilities, and self-check against provided answers. Follow with pair shares for tricky cases.

Analyze how tree diagrams visually represent conditional probabilities.

Facilitation Tip

For Individual: Spinner Dependency Challenges, have students trade spinner sheets with a partner to verify each other’s branch labels before calculating.

What to look for

Present students with a scenario, such as drawing two marbles from a bag without replacement. Ask them to draw the first two levels of the tree diagram, labeling each branch with the correct probability. Check for accurate initial probabilities and conditional probabilities on the second level.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with a quick physical model like drawing two cards from a small deck to show how the first card changes the deck for the second draw. Use think-alouds to model updating probabilities on the fly. Avoid rushing to the formula; let students struggle with the drawing first, then guide them to see why multiplication and addition work the way they do.

By the end of these activities, students will draw accurate tree diagrams for two-step dependent events, label second-level branches with correct conditional probabilities, and use the diagram to calculate combined probabilities without mixing up multiplication and addition. They will explain in their own words why each branch probability changes after the first event.

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Watch Out for These Misconceptions

• During Small Groups: Bead Bag Simulations, watch for students who add probabilities at each draw instead of updating the total and recalculating the remaining counts.

  Have the group pause after the first draw, recount the beads left in the bag, and redraw the second level of the tree to show the new totals before labeling probabilities.

• During Pairs: Diagnostic Test Trees, watch for students who use the original probabilities for both events, treating them as independent.

  Ask one partner to explain the path while the other simulates a draw; when the simulation outcome contradicts the original probability, the group redraws the second level with updated conditionals.

• During Whole Class: Chain Event Build, watch for students who add the probabilities along a single path instead of multiplying.

  Prompt students to test their method by running a quick experiment with 20 trials and compare the experimental frequency to the product of the two probabilities along their chosen path.

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Methods used in this brief

• Collaborative Problem-Solving