🎯 Learning Objectives
- write sample spaces systematically;
- calculate probabilities of single events;
- distinguish favourable from total outcomes;
- avoid assuming unequal outcomes are equally likely.
Introduction
This lesson develops Sample Spaces and Single Events 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.
Theoretical Probability
Worked Examples
Fair die sample space S={1,2,3,4,5,6}.
Event A: number greater than 4 = {5,6}.
P(A)=2/6=1/3.
A bag contains 3 red and 5 blue identical counters.
P(red)=3/(3+5)=3/8.
Systematic Listing
Write outcomes in an organised order so none are omitted or repeated.
Practice Questions
- Find P(odd) on a fair die.
- A letter is selected from MATHS. Find P(A).
- A spinner has equal sectors R,R,B,G. Find P(R).
- Why can favourable/total be used only when outcomes are equally likely?
✅ Check Your Work
Show answers and working
- Odd outcomes={1,3,5}; P=3/6=1/2.
- There are 5 letters and one A, so P(A)=1/5.
- Two of four sectors are red, so P(R)=2/4=1/2.
- Counting outcomes treats each outcome as having the same probability; this gives a wrong answer when chances differ.
🧪 Math Lab: Build a Sample Space
Choose an experiment and reveal every possible outcome.
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.