Probability of Combined Events (Dependent)

Active learning helps students grasp dependent probability by making abstract sample space changes visible through concrete actions. Physical draws and real-time tallies transform conditional formulas from symbols into evidence they can see and question.

• Secondary 3
• Mathematics
• 4 activities
• 30 min–45 min

Learning Objectives

1. Calculate the probability of two dependent events occurring in sequence using the formula P(A and B) = P(A) × P(B|A).
2. Construct and interpret tree diagrams to represent scenarios involving dependent events and conditional probabilities.
3. Analyze how the removal of an item from a sample space affects the probability of subsequent events.
4. Compare the calculated probabilities of dependent events with outcomes from simulated trials.

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Card Draw Simulation: Without Replacement

Provide decks of cards to small groups. Students draw two cards without replacement, record outcomes like both aces or both hearts over 20 trials, then calculate experimental probabilities. Compare results to theoretical values from tree diagrams drawn as a class.

Prepare & details

Explain how the concept of 'without replacement' changes the structure of a probability model.

Facilitation Tip: For the Card Draw Simulation, have students draw and tally results as a class before calculating theoretical probabilities to highlight the gap between expectation and outcome.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Collaborative Problem-Solving

⏱ 30 min·Pairs

Tree Diagram Build: Sampling Scenarios

Pairs receive scenarios such as drawing marbles from a bag without replacement. They construct tree diagrams showing probabilities at each branch, label conditional probabilities, and compute overall P(A and B). Share and critique with the class.

Prepare & details

Analyze how the probability of the second event is affected by the outcome of the first event.

Facilitation Tip: During Tree Diagram Build, insist each group labels branches with updated denominators and conditional phrases such as 'given the first was...'.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Collaborative Problem-Solving

⏱ 45 min·Whole Class

Class Survey Relay: Dependent Choices

Whole class participates in a relay where students sequentially pick names from a hat without replacement for roles, recording probabilities. Groups update a shared tree diagram after each draw and predict next outcomes.

Prepare & details

Construct a tree diagram for a scenario involving dependent events.

Facilitation Tip: In the Class Survey Relay, use a visible tally board so the shrinking pool of choices is obvious to every student.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Collaborative Problem-Solving

⏱ 40 min·Small Groups

Marble Probability Stations

Set up stations with bags of colored marbles. Groups draw two without replacement at each station, tally results on charts, and derive Pboth same color. Rotate stations to test different ratios.

Prepare & details

Explain how the concept of 'without replacement' changes the structure of a probability model.

Facilitation Tip: At Marble Probability Stations, rotate groups every 4 minutes so everyone experiences multiple dependent draws and sees consistent pattern shifts.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

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

Teaching This Topic

Teachers should begin with physical simulations to build intuition, then transition to tree diagrams as a visual model that tracks changing probabilities. Avoid rushing to the formula P(A and B) = P(A) × P(B|A) before students see why the denominator changes. Research shows that students who construct their own tree diagrams from simulation data internalize conditional probability more deeply than those who only receive pre-made diagrams.

What to Expect

Students will explain how the first outcome reduces the sample space and adjust the second probability accordingly. They will correctly use the formula P(A and B) = P(A) × P(B|A) and defend their reasoning with tree diagrams or frequency data from simulations.

Watch Out for These Misconceptions

Common Misconception

During Card Draw Simulation: Watch for students who assume the second draw’s probability remains 26/52 for a heart regardless of the first card drawn.

What to Teach Instead

After students draw and tally results, ask them to recalculate the second probability based on the new deck size and compare it with their tallies to correct the misconception.

Common Misconception

During Tree Diagram Build: Watch for students who multiply original probabilities P(A) and P(B) directly for dependent events.

What to Teach Instead

Have students test their diagrams against the Card Draw Simulation results, prompting them to adjust branch probabilities and see why P(B|A) ≠ P(B).

Common Misconception

During Tree Diagram Build: Watch for students who draw independent-style tree branches that do not reflect changing sample spaces.

What to Teach Instead

Ask groups to present their diagrams to the class and justify each branch’s probability using updated denominators, using peer feedback to correct errors.

Assessment Ideas

Quick Check

After Card Draw Simulation, present students with a new scenario: 'A deck has 4 Aces and 48 non-Aces. Two cards are drawn without replacement. What is the probability both are Aces?' Have students write the formula and final answer on a mini whiteboard and hold it up for immediate feedback.

Discussion Prompt

During Class Survey Relay, pose the question: 'If your class has 12 boys and 18 girls, how does the probability of selecting a girl as the second person change if the first selection was a boy? Discuss how the pool shrinks and why conditional probability matters here.'

Exit Ticket

After Marble Probability Stations, give students a partially completed tree diagram for drawing two marbles from a bag of 5 red and 3 green marbles without replacement. Ask them to label the final branch with the probability of drawing red followed by green and explain their calculation in one sentence.

Extensions & Scaffolding

• Challenge: Ask students to design a dependent probability game with unfair odds and have peers calculate the true winning chances.
• Scaffolding: Provide partially completed tree diagrams with missing branch labels or probabilities for students to fill in before calculating final values.
• Deeper exploration: Have students compare dependent probabilities in sampling without replacement to sampling with replacement, then explain when the two scenarios yield similar results.

Key Vocabulary

Dependent Events	Two or more events where the outcome of one event affects the probability of the others.
Conditional Probability	The probability of an event occurring, given that another event has already occurred. Denoted as P(B|A).
Without Replacement	A scenario where an item, once selected or removed, is not returned to the sample space, thus changing the probabilities for subsequent selections.
Tree Diagram	A graphical tool used to display all possible outcomes of a sequence of events, showing probabilities at each branch.