Mutually exclusive
Cannot happen together in one trial.
🎯 Learning Objectives
- identify independent events;
- apply the multiplication rule;
- distinguish independence from mutual exclusivity;
- calculate repeated-event probabilities.
Introduction
This lesson develops Independent 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.
Independent Events
Events are independent when one event happening does not change the probability of the other.
For independent A and B: P(A and B)=P(A)×P(B).
Worked Examples
Two fair coin tosses:
P(HH)=1/2×1/2=1/4.
Roll a fair die and toss a fair coin:
P(6 and H)=1/6×1/2=1/12.
Do Not Confuse These Ideas
Independent
One does not change the probability of the other.
Two non-impossible mutually exclusive events are not independent: if one happens, the other becomes impossible.
Practice Questions
- Find P(three heads in three fair tosses).
- Find P(rolling an even number and tossing a tail).
- Explain why two separate die rolls are independent.
✅ Check Your Work
Show answers and working
- (1/2)³=1/8.
- P(even)=3/6=1/2; P(tail)=1/2; product=1/4.
- The first roll does not alter the probabilities on the second fair roll.
🧪 Math Lab: Independent Events
Change two independent probabilities and multiply them.
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.