🎯 Learning Objectives
- adjust probabilities after an item is not replaced;
- construct changing-probability trees;
- calculate dependent-event probabilities;
- check remaining totals carefully.
Introduction
This lesson develops Probability Without 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.
Without Replacement
A bag contains 4 red and 3 blue counters.
P(red then red)=4/7×3/6=12/42=2/7.
P(red then blue)=4/7×3/6=2/7.
P(blue then red)=3/7×4/6=2/7.
Dependence
The second probability depends on the first selection because the composition of the bag changes.
Practice Questions
- Find P(two blue).
- Find P(one of each).
- Why is the second denominator 6 rather than 7?
- What would change if the first counter were replaced?
✅ Check Your Work
Show answers and working
- 3/7×2/6=6/42=1/7.
- P(RB)+P(BR)=2/7+2/7=4/7.
- Only six counters remain after the first selection.
- The bag would return to 4 red and 3 blue, so second probabilities would again use denominator 7.
🧪 Math Lab: Without-Replacement Bag
A bag has 3 blue and 2 red balls. Compare the second-draw probabilities after each first draw.
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.