🎯 Learning Objectives
- construct tree diagrams;
- label branches with probabilities;
- multiply along branches and add relevant outcomes;
- solve with-replacement problems.
Introduction
This lesson develops Tree Diagrams with Replacement from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Tree-Diagram Rules
Worked Example
A bag contains 3 red and 2 blue counters. One is selected, replaced, and another selected.
P(two red)=3/5×3/5=9/25.
P(one of each)=3/5×2/5 + 2/5×3/5=12/25.
Practice Questions
- Find P(two blue) in the example.
- Find P(at least one red).
- Why do second-stage probabilities stay unchanged?
✅ Check Your Work
Show answers and working
- 2/5×2/5=4/25.
- 1−P(two blue)=1−4/25=21/25.
- The selected counter is replaced, restoring the original bag composition.
🧪 Math Lab: With-Replacement Tree
A bag has 3 blue and 2 red balls. Draw, replace, then draw again.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.