GCSE MATHS • HIGHER TIER
Independent Events Made Simple
Recognise independent events and calculate multi-step probabilities using the multiplication rule.
Clear method
Exam practice
Worked proofs
Independent Events in Four Moves
1
Identify
Decide whether one event affects the probability of the next event.
2
Find
Write the probability of each required event.
3
Multiply
Multiply probabilities for events that must both happen.
4
Check
Confirm that the events are independent and simplify the answer.
What You’ll Learn
Understand what independent events are.
Recognise common independent-event situations.
Use the multiplication rule correctly.
Calculate probabilities for repeated coin tosses and dice rolls.
Understand why replacement can create independent events.
Use tree diagrams to organise independent probabilities.
Distinguish independent events from dependent events.
Key Rules
For independent events, probabilities are multiplied together.
Probability of Event A and Event B = Probability of A × Probability of B
Worked Proof: Independent Events
Step-by-Step Method
- 1. Decide whether the events are independent.
- 2. Write the probability of each required event separately.
- 3. If an item is selected from a bag, check whether it is replaced.
- 4. When both independent events must happen, multiply their probabilities.
- 5. If there are three or more independent events, multiply all the required probabilities.
More Worked Examples
1
Question: What is the probability of getting two heads when tossing a fair coin twice?
Probability of heads = 1/2.
Probability of two heads = 1/2 × 1/2
= 1/4.
= 1/4.
2
Question: What is the probability of rolling a 6 and then another 6 on a fair dice?
Probability = 1/6 × 1/6
= 1/36.
3
Question: What is the probability of getting a head and then a tail when tossing a coin twice?
Probability = 1/2 × 1/2
= 1/4.
4
Question: A spinner has 5 equal sections and 2 are red. What is the probability of landing on red twice?
Probability = 2/5 × 2/5
= 4/25.
- Common GCSE Mistakes
- Adding probabilities when they should be multiplied.
- Forgetting that replacement is required in some situations.
- Confusing independent and dependent events.
- Changing a probability even though the event is independent.
- Exam Tips
- Check first whether the events affect each other.
- Look carefully for the word replacement.
- Multiply probabilities when both independent events must occur.
- Use a tree diagram for multi-step questions when helpful.
- Do not treat without-replacement selections as independent.
Skills Used in Independent Events
Probability
Independent Events
Multiplication of Fractions
Replacement
Tree Diagrams
Dice and Coin Probabilities
Ready to Practise?
Printable Worksheet
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Interactive Quiz
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Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Independent Events Tutorial
Step-by-step walkthroughs of Independent Events with clear methods and exam tips.
Frequently Asked Questions
They are events where the outcome of one event does not affect the outcome of another.
Multiply the probabilities when the required independent events must both happen.
Yes. The result of one fair coin toss does not affect the next toss.
Replacing an item restores the original probabilities, so repeated selections can remain independent.
The probabilities can change, making the events dependent.
Yes. They help organise outcomes, and probabilities along branches are multiplied.