EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Mutually exclusive events
Probability · Lesson 2 of 6
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. What is the probability of an impossible event?
0
2. What is the probability of a certain event?
1
3. Work out 1 − 0.35.
0.65
4. Simplify 10/25.
2/5
5. Work out 3/10 + 2/10.
½
Learning Objectives
1. Use the probability scale from 0 to 1.
2. Recognise mutually exclusive events.
3. Use P(A or B) = P(A) + P(B) for mutually exclusive events.
4. Use P(not A) = 1 − P(A) and complete probability tables.
The Probability Scale
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Every probability lies between 0 and 1. |
A probability scale from 0 to 1 with examples of impossible, unlikely, even chance, likely and certain events.
Key Facts
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Range Every probability is between 0 and 1, including 0 and 1. |
Total The probabilities of all the possible outcomes add up to 1. |
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Not happening P(not A) = 1 − P(A). |
Mutually exclusive Events that cannot both happen at the same time, such as rolling a 2 and rolling a 5 on one dice. |
The Addition Rule
For mutually exclusive events, P(A or B) = P(A) + P(B).
This only works when A and B cannot happen together.
Finding a Missing Probability
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A biased spinner has probabilities: red 0.3, blue 0.5, green x. Find x, and P(red or blue). |
1. The probabilities add up to 1
0.3 + 0.5 + x = 1
2. Solve
x = 0.2
3. Red and blue are mutually exclusive, so add
0.3 + 0.5 = 0.8
Answer: x = 0.2 and P(red or blue) = 0.8.
A Probability Table
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A four-sided spinner has P(1) = 0.1, P(2) = 0.25, P(3) = x and P(4) = 0.3. Find x and P(not 4). |
1. Probabilities add up to 1
0.1 + 0.25 + x + 0.3 = 1
2. Solve
0.65 + x = 1, so x = 0.35
3. Not 4
1 − 0.3 = 0.7
Answer: x = 0.35 and P(not 4) = 0.7.
Counters in a Bag
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A bag has 5 red, 3 blue and 2 green counters. One counter is taken at random. Find the probability that it is red or green. |
1. Total counters
5 + 3 + 2 = 10
2. Red and green cannot both be taken
Mutually exclusive
3. Add the probabilities
5/10 + 2/10 = 7/10
Answer: 7/10
HIGHER TIER
Events That Can Overlap
When A and B can both happen.
Not Mutually Exclusive HIGHER
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P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B). |
1. Adding counts the overlap twice
0.5 + 0.4 = 0.9
2. Subtract the overlap once
0.9 − 0.2
3. Work it out
0.7
Answer: P(A or B) = 0.7
Key Terms
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Mutually exclusive Events that cannot happen at the same time. |
Probability scale A line from 0 (impossible) to 1 (certain). |
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Complement The event "not A", with probability 1 − P(A). |
Exhaustive A set of events that covers every possible outcome. |
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Biased Not fair; outcomes are not equally likely. |
Random Every item has the same chance of being chosen. |
Your Task: Complete the Table
10 minutes
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A biased dice has probabilities: 1: 0.1, 2: 0.15, 3: 0.2, 4: 0.25, 5: x, 6: 2x. Find x, then find the probability of an even number and the probability of not rolling a 6. 1. Use "add to 1". 2. Add for "or". 3. Subtract from 1 for "not". |
A good answer shows: 0.1 + 0.15 + 0.2 + 0.25 + 3x = 1, so 3x = 0.3 and x = 0.1. P(6) = 0.2. Even: 0.15 + 0.25 + 0.2 = 0.6. Not 6: 0.8.
Note: Point out that even numbers are mutually exclusive, so the probabilities add.
Can I...?
☐ Place an event on the probability scale.
☐ Use "add up to 1" to find a missing probability.
☐ Use 1 − P(A).
☐ Recognise mutually exclusive events.
☐ Add probabilities for "or".
☐ Complete a probability table.
☐ Work with counters in a bag.
☐ Use P(A) + P(B) − P(A and B) (Higher).
Summary
✓ All the probabilities add up to 1.
✓ P(not A) = 1 − P(A).
✓ Mutually exclusive: P(A or B) = P(A) + P(B).
✓ Higher: subtract the overlap when events can both happen.
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EXAM FOCUS A bag has 5 red, 3 blue and 2 green counters. A counter is taken at random. Work out the probability that it is red or green. (2 marks) Red and green cannot both happen, so add the two probabilities. The total number of counters is the denominator. |
Exam Practice: Mutually Exclusive Events
Answer all questions. Show your working. · 25 minutes
▸ Question 1 · 2 marks · Non-calculator. A biased spinner lands on red with probability 0.3, on blue with probability 0.5, and otherwise on green. Work out the probability that the…
▸ Question 2 · 2 marks · Non-calculator. The probability that it rains tomorrow is 0.35. Work out the probability that it does not rain tomorrow.
▸ Question 3 · 3 marks · Non-calculator. A four-sided spinner is biased. The probability of each score is shown: score 1: 0.1, score 2: 0.25, score 3: x, score 4: 0.3. Work out the…
▸ Question 4 · 2 marks · Non-calculator. A bag has 5 red counters, 3 blue counters and 2 green counters. One counter is taken at random. Work out the probability that it is red or…
▸ Question 5 · 3 marks · Non-calculator. In a class of 30 students, 12 play football, 9 play tennis and 5 play both. Explain why the events "plays football" and "plays tennis" are…
▸ Question 6 · 3 marks · Non-calculator. A and B are two events. P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Work out P(A or B).
Question 1 · 2 marks · Non-calculator
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“A biased spinner lands on red with probability 0.3, on blue with probability 0.5, and otherwise on green. Work out the probability that the spinner lands on green.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ 1 − 0.3 − 0.5. M1
▸ 0.2. A1
▸ Model answer. 1 − 0.3 − 0.5 = 0.2.
Question 2 · 2 marks · Non-calculator
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“The probability that it rains tomorrow is 0.35. Work out the probability that it does not rain tomorrow.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ 1 − 0.35. M1
▸ 0.65. A1
▸ Model answer. 1 − 0.35 = 0.65.
Question 3 · 3 marks · Non-calculator
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“A four-sided spinner is biased. The probability of each score is shown: score 1: 0.1, score 2: 0.25, score 3: x, score 4: 0.3. Work out the value of x, and the probability of scoring 2 or 4.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 3 · mark scheme
3 marks available. Award a mark for each point made.
▸ Probabilities sum to 1. M1
▸ x = 0.35. A1
▸ 0.55. A1
▸ Model answer. 0.1 + 0.25 + x + 0.3 = 1, so x = 0.35. P(2 or 4) = 0.25 + 0.3 = 0.55.
Question 4 · 2 marks · Non-calculator
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“A bag has 5 red counters, 3 blue counters and 2 green counters. One counter is taken at random. Work out the probability that it is red or green.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ Red and green probabilities added. M1
▸ 7/10. A1
▸ Model answer. 5/10 + 2/10 = 7/10.
Question 5 · 3 marks · Non-calculator
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“In a class of 30 students, 12 play football, 9 play tennis and 5 play both. Explain why the events "plays football" and "plays tennis" are not mutually exclusive.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 5 · mark scheme
3 marks available. Award a mark for each point made.
▸ Reference to the 5 who play both. M1
▸ Both events can happen together. C1
▸ A clear conclusion. C1
▸ Model answer. Five students play both, so the two events can happen at the same time, and they are not mutually exclusive.
Question 6 · 3 marks · Non-calculator
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“A and B are two events. P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Work out P(A or B).” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ 0.5 + 0.4. M1
▸ Subtracting the overlap. M1
▸ 0.7. A1
▸ Model answer. P(A or B) = 0.5 + 0.4 − 0.2 = 0.7.