Tree Diagrams for Dependent Events

Active tasks let students see how probabilities shift when items are removed, turning abstract fractions into concrete evidence. Moving from cards to marbles makes the shrinking pool visible, so students trust the updated branch fractions on their diagrams.

• Year 10
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

1. Calculate the probability of sequential dependent events using a tree diagram.
2. Compare the structure and probability calculations for independent versus dependent events represented by tree diagrams.
3. Explain how the removal of an item affects the probability of subsequent events in 'without replacement' scenarios.
4. Design a word problem that requires the use of a tree diagram to solve for dependent probabilities.
5. Analyze the outcomes of a tree diagram to determine the likelihood of specific compound events.

Decision Matrix

⏱ 35 min·Small Groups

Hands-On: Card Simulation Stations

Provide decks of cards at stations. Groups draw two or three cards without replacement, record outcomes on mini whiteboards, and construct tree diagrams for probabilities like both red or first ace then king. Compare group results to class predictions. Rotate stations for variety.

Prepare & details

Explain how the concept of 'without replacement' alters probabilities in subsequent events.

Facilitation Tip: During Card Simulation Stations, have each pair record the remaining cards after every draw so they can immediately see why the next branch fraction must be smaller.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 25 min·Pairs

Pairs: Custom Problem Swap

Pairs design a 'without replacement' scenario, such as coloured marbles from a bag, and draw its tree diagram with probabilities. Swap problems with another pair, solve using trees, then discuss solutions and check calculations together.

Prepare & details

Compare the structure of tree diagrams for independent versus dependent events.

Facilitation Tip: While pairs swap problems in Custom Problem Swap, circulate and ask one partner to explain the updated branch fractions to the other before they calculate.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 30 min·Whole Class

Whole Class: Tree Diagram Relay

Divide class into teams. Project a multi-stage dependent event. One student per team adds a branch to the shared board tree with updated probability, next teammate multiplies or adds as needed. First accurate tree wins.

Prepare & details

Design a problem where a tree diagram is essential for understanding dependent probabilities.

Facilitation Tip: In Tree Diagram Relay, stop each group after the first branch and ask them to justify their updated fraction with the cards or marbles they have left in front of them.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Decision Matrix

⏱ 20 min·Individual

Individual: Marble Bag Trials

Each student gets a bag of coloured marbles, performs 20 draws without replacement for pairs of colours, tallies results, and builds a tree diagram to predict theoretical probabilities. Share findings in plenary.

Prepare & details

Explain how the concept of 'without replacement' alters probabilities in subsequent events.

Facilitation Tip: For Marble Bag Trials, ask students to sketch the first two branches on mini whiteboards before they pick any marbles, forcing them to predict the changing totals.

Setup: Groups at tables with matrix worksheets

Materials: Decision matrix template, Option description cards, Criteria weighting guide, Presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management

Teaching This Topic

Start with a physical model—cards or marbles—so students feel the pool shrink before they draw the tree. Ask them to predict what the next branch fraction should be, then confirm it with the remaining items. Avoid rushing to the abstract diagram; let the evidence from the removal shape the fractions on the branches.

What to Expect

You will hear students explain why probabilities change after each draw, watch them label branches correctly with updated fractions, and see them multiply along paths then add across paths without mixing the two steps.

Watch Out for These Misconceptions

Common Misconception

During Card Simulation Stations, watch for students who keep the same branch fractions across all draws because they assume the deck never changes.

What to Teach Instead

Have them stop after the first draw, count the cards left in their hand, and recalculate the next branch fraction before they continue drawing. Ask them to write the updated fraction on the board for the group to see.

Common Misconception

During Custom Problem Swap, watch for partners who multiply probabilities across different branches rather than along a single path.

What to Teach Instead

Tell them to trace one colored path with their finger from start to end and label each branch fraction before they multiply; if they try to multiply from different branches, hand them a red pen and have them cross it out.

Common Misconception

During Tree Diagram Relay, watch for groups who skip updating the second branch because they believe the event is still independent.

What to Teach Instead

Freeze the relay and ask each group to hold up the remaining cards or marbles; then require them to write the new denominator on the board before they proceed to the next step.

Assessment Ideas

Quick Check

After Card Simulation Stations, give each pair a quick scenario with two dependent draws and ask them to draw the tree and calculate the final probability within five minutes. Collect the diagrams to check branch values and path multiplication.

Discussion Prompt

During Marble Bag Trials, ask students to turn to a neighbor and explain how the calculation changes if the marbles were drawn with replacement instead. Have two pairs share their explanations with the class.

Exit Ticket

After Tree Diagram Relay, hand out partially completed trees for a two-draw dependent event. Students fill in the missing branch fractions and calculate the probability of one specific outcome before leaving the room.

Extensions & Scaffolding

• Challenge students who finish early to design a new scenario with three dependent events and build the corresponding tree, explaining how each branch updates.
• For students who struggle, provide bags with only three marbles total and have them complete one full path before adding the second branch.
• Deeper exploration: give a scenario with unequal initial counts (e.g., 5 red, 1 blue) and ask students to find the smallest initial count that makes the probability of two blues at least 0.10.

Key Vocabulary

Dependent Events	Events where the outcome of the first event affects the probability of the second event occurring.
Without Replacement	A condition in probability where an item, once selected, is not returned to the sample space, thus changing the probabilities for subsequent selections.
Conditional Probability	The probability of an event occurring, given that another event has already occurred.
Branch Probability	The probability assigned to each path or outcome on a tree diagram, which may change for dependent events.