Probability trees are a core skill in IB Mathematics: Applications and Interpretation (AI). Whether you're preparing for Paper 1 or Paper 2 at Standard Level (SL) or Higher Level (HL), knowing how to construct and interpret probability tree diagrams is essential.
This guide will walk you through how to use probability trees in IB Math AI, with simple steps, exam-style examples, and common mistakes to avoid—making even the trickiest questions manageable.
What Is a Probability Tree?
A probability tree is a diagram that helps visualize and calculate the probability of multiple events occurring in sequence. Each branch represents an outcome, and each path shows a possible combination of outcomes.
You use it when:
- There are two or more stages in a probability situation.
- The outcome of one event affects the next (conditional probability).
- You need to add and multiply probabilities systematically.
📊 Think of it as a visual roadmap to multi-step probability problems.
When Are Probability Trees Used in IB Math AI?
You’ll encounter probability trees in:
- Binomial events (e.g., win or lose, success or fail)
- Conditional probability questions
- Problems involving replacement and no replacement
- Real-world modeling (e.g., surveys, game scenarios, genetics)
These often appear in Paper 1 (no calculator) for theory-focused questions, and Paper 2 or 3 (calculator allowed) when interpretation and modeling are emphasized.
Components of a Probability Tree Diagram
Each probability tree consists of:
- Nodes: Points where branches split (each event stage)
- Branches: Lines representing outcomes, labeled with their probabilities
- Paths: A complete sequence from start to end
- Outcomes: Endpoints, often used to find combined or conditional probability
💡 Always check that each set of branches adds up to 1, which ensures the diagram is logically sound.
Step-by-Step: How to Build a Probability Tree
Let’s walk through a simple example:
Scenario:
A bag has 3 red balls and 2 blue balls. One ball is drawn, not replaced, and then another is drawn.
Step 1: Identify the events
- First draw: Red or Blue
- Second draw (depends on first): Red or Blue
Step 2: Draw branches for the first event
- P(Red) = 3/5
- P(Blue) = 2/5
Step 3: Add second-stage branches
- If Red was drawn: now 2 red, 2 blue → P(Red) = 2/4, P(Blue) = 2/4
- If Blue was drawn: still 3 red, 1 blue → P(Red) = 3/4, P(Blue) = 1/4
Step 4: Multiply along each path
- P(Red then Red) = (3/5) × (2/4) = 6/20
- P(Red then Blue) = (3/5) × (2/4) = 6/20
- P(Blue then Red) = (2/5) × (3/4) = 6/20
- P(Blue then Blue) = (2/5) × (1/4) = 2/20
Step 5: Add paths if needed
For example, P(exactly one red) = P(Red then Blue) + P(Blue then Red) = 6/20 + 6/20 = 12/20
How to Calculate Combined and Conditional Probabilities
- Combined probabilities (AND): Multiply along a path.
- e.g., P(A then B) = P(A) × P(B|A)
- Conditional probabilities (GIVEN): Use relevant paths.
- e.g., P(A|B) = P(A and B) / P(B)
📘 Always read carefully whether the event is dependent or independent—this changes the branches.
Calculator Tips for IB Exams
If you're in Paper 2 or 3, use your GDC for faster results:
- Use lists or matrices to store probabilities
- Use tree diagram apps or probability functions
- For conditional calculations, store results as variables for re-use
⚠️ But don't skip showing your work—examiners need to see logic, even if the calculator gets the answer.
Common Mistakes to Avoid with Probability Trees
- Probabilities not summing to 1 at each stage
- Forgetting to adjust probabilities after a non-replacement
- Confusing independent and dependent events
- Skipping steps in multiplication or addition
- Failing to label the outcomes clearly
✅ Always double-check that each outcome path is complete and accurately calculated.
Practice Question with Full Solution
Question:
A biased coin has a 0.6 chance of landing heads. It is tossed twice. What is the probability of getting exactly one head?
Step 1: Draw Tree
- First toss: H (0.6), T (0.4)
- Second toss: H (0.6), T (0.4)
Step 2: Paths for exactly one head
- H then T: 0.6 × 0.4 = 0.24
- T then H: 0.4 × 0.6 = 0.24
Step 3: Add paths
P(Exactly one head) = 0.24 + 0.24 = 0.48
FAQs About Probability Trees in IB Math AI
1. Are probability trees required for HL as well?
Yes. They're used in HL too, especially in real-world applications and Paper 3 questions.
2. What if events are independent?
The branches for the second event won’t change based on the first—use same probabilities for all second-stage branches.
3. Do I need to draw trees in calculator papers?
Not always, but drawing helps with clarity and partial marks, even if you use the GDC for calculations.
4. How can I practice these questions?
Use RevisionDojo’s topic-based practice packs and past IB exams that focus on conditional and compound probabilities.
5. Is tree diagram use tested directly in exams?
Sometimes, but often it's hidden in a multi-part question. You'll need to know how to apply it, even if not explicitly asked.
6. Can tree diagrams be used in the IA?
Yes—especially if your exploration involves sequential probability, decision-making models, or game theory.
Conclusion: Mastering Probability Trees with Confidence
Understanding how to use probability trees in IB Math AI unlocks your ability to solve complex, multi-step problems with ease. They're powerful, visual, and conceptually rich tools that show up in both SL and HL exams.
Start by practicing simple examples. Then move on to layered, real-world applications. With time, these diagrams will become second nature—and your scores will reflect that mastery.
Further Resources for IB Math AI Students
Need more practice?
👉 Visit RevisionDojo’s IB Math AI Blog for:
- Topic-by-topic walkthroughs
- Tree diagram exercises
- Paper 2 strategy guides
- Calculator function tutorials
