🎯 Learning Objectives
- use complementary events;
- find missing probabilities;
- apply the total-probability rule;
- check that complete outcome probabilities sum to 1.
Introduction
This lesson develops Complements and Basic Probability Rules 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.
Complement Rule
If P(rain)=0.35, then P(no rain)=1−0.35=0.65.
Complete Outcome Sets
If the only possible outcomes are red, blue and green, their probabilities add to 1.
P(red)=0.25 and P(blue)=0.40.
P(green)=1−0.25−0.40=0.35.
“At Least One”
It is often easier to calculate the complement:
Practice Questions
- P(A)=0.72. Find P(not A).
- P(red)=1/5, P(blue)=1/2. Find P(green), assuming these are the only colours.
- Explain why branch probabilities leaving one tree node total 1.
✅ Check Your Work
Show answers and working
- 1−0.72=0.28.
- 1−1/5−1/2=1−2/10−5/10=3/10.
- The branches list all mutually exclusive next outcomes, so one of them must happen.
🧪 Math Lab: Complement Finder
Enter the probability of an event to find the probability that it does not happen.
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.