EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Venn diagrams and set notation
Probability · Lesson 6 of 6
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. List the even numbers from 1 to 10.
2, 4, 6, 8, 10
2. List the multiples of 3 from 1 to 10.
3, 6, 9
3. What number is in both lists above?
6
4. Work out 40 − 29.
11
5. Simplify 8/16.
½
Learning Objectives
1. Use set notation: ∈, ∪, ∩, A', xi and n(A).
2. Draw and complete a Venn diagram.
3. Shade the region for a set expression.
4. Find probabilities from a Venn diagram, including conditional ones (Higher).
Set Notation
Learn these symbols.
|
Symbol |
Meaning |
Example |
|---|---|---|
|
xi |
The universal set: everything being considered |
xi= {1, 2, ..., 10} |
|
A ∪ B |
Union: in A or B or both |
{2, 3, 4, 6, 8, 9, 10} |
|
A ∩ B |
Intersection: in both A and B |
{6} |
|
A' |
Complement: not in A |
{1, 3, 5, 7, 9} |
|
n(A) |
The number of elements in A |
n(A) = 5 |
|
∈ |
"is an element of" |
4 ∈A |
Shading Set Regions
|
Shade the region that matches the notation. |
Four Venn diagrams showing the shaded regions for A union B, A intersection B, the complement of A, and the complement of A intersected with B.
Sets of Numbers
|
xi= {1, 2, 3, ..., 10}, A = the even numbers and B = the multiples of 3. List A, B, A ∩ B, A ∪ B and A'. |
1. A
{2, 4, 6, 8, 10}
2. B
{3, 6, 9}
3. In both
A ∩ B = {6}
4. In either
A ∪ B = {2, 3, 4, 6, 8, 9, 10}
5. Not in A
A' = {1, 3, 5, 7, 9}
Answer: See the lists above; n(A ∪ B) = 7.
Completing a Venn Diagram
Start in the middle and work outwards.
|
1 Fill in the overlap first The number in both sets |
2 Work out the rest of each set Subtract the overlap from each total |
3 Fill the outside Universal total minus everything inside the circles |
4 Check All the numbers add up to the universal total |
A Survey
|
40 students were asked about music (M) and art (R). 22 take music, 15 take art and 8 take both. Complete the Venn diagram and find the number who take neither. |
1. Both
8
2. Music only
22 − 8 = 14
3. Art only
15 − 8 = 7
4. Inside the circles
14 + 8 + 7 = 29
5. Neither
40 − 29 = 11
Answer: Music only 14, both 8, art only 7, neither 11.
Probabilities from a Venn Diagram
|
For the survey, a student is chosen at random. Find (a) P(neither), (b) P(M ∪ R). |
1. (a) Neither
11/40
2. (b) Music or art (or both)
14 + 8 + 7 = 29, so 29/40
Answer: (a) 11/40 (b) 29/40
HIGHER TIER
Conditional Probability from a Venn Diagram
Restrict to the given set.
Given That HIGHER
|
For the survey, find P(M midR): the probability a student takes music, given that they take art. |
1. "Given that they take art" restricts to the art circle
15 students
2. Of these, the number who also take music
8
3. Probability
8/15
Answer: 8/15
Key Terms
|
Set A collection of items. |
Element One item in a set. |
|
Universal set The set of everything being considered, xi. |
Union Items in one set or the other or both, ∪. |
|
Intersection Items in both sets, ∩. |
Complement Items not in the set, A'. |
Your Task: Class Survey
12 minutes
|
In a class of 30 students, 18 have a dog, 12 have a cat and 5 have both. Draw a Venn diagram, and find the probability that a randomly chosen student has (a) a dog only (b) neither (c) a cat given that they have a dog. 1. Fill in the overlap first. 2. Subtract for the other regions. 3. Use the right total. |
A good answer shows: Dog only 18 − 5 = 13, cat only 12 − 5 = 7, both 5, neither 30 − 25 = 5. (a) 13/30 (b) 5/30 = 1/6 (c) 5/18.
Note: Point out that part (c) restricts the total to the 18 dog owners.
Can I...?
☐ Use the set notation symbols.
☐ List the elements of A ∩ B and A ∪ B.
☐ Find A'.
☐ Fill in a Venn diagram from totals.
☐ Shade the correct region.
☐ Find a probability from a Venn diagram.
☐ Find P(A midB) (Higher).
☐ Check the total.
Summary
✓ A ∪ B: in A or B. A ∩ B: in both. A': not in A.
✓ Fill the overlap first, then each circle, then the outside.
✓ The numbers must add up to the universal total.
✓ Higher: P(A midB) uses B as the new total.
|
EXAM FOCUS 40 students were asked about music and art. 22 take music, 15 take art and 8 take both. Draw a Venn diagram to show this information. (4 marks) Start with the overlap (8), then subtract it from each circle total. Finish with the number outside. |
Exam Practice: Venn Diagrams and Set Notation
Answer all questions. Show your working. · 30 minutes
▸ Question 1 · 2 marks · Non-calculator. xi= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {even numbers} and B = {multiples of 3}. List the members of A ∩ B.
▸ Question 2 · 2 marks · Non-calculator. Using the same sets, list the members of A'.
▸ Question 3 · 4 marks · Non-calculator. 40 students were asked whether they take music (M) and art (R). 22 take music, 15 take art and 8 take both. Complete the Venn diagram.
▸ Question 4 · 2 marks · Non-calculator. One of the 40 students is chosen at random. Use your Venn diagram to work out the probability that the student takes neither music nor art.
▸ Question 5 · 2 marks · Non-calculator. Write down the number of students in M ∪ R.
▸ Question 6 · 3 marks · Non-calculator. One of the 40 students is chosen at random. Given that the student takes art, work out the probability that they also take music.
Question 1 · 2 marks · Non-calculator
|
“xi= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {even numbers} and B = {multiples of 3}. List the members of A ∩ B.” |
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.
▸ A = {2, 4, 6, 8, 10} and B = {3, 6, 9}. M1
▸ {6}. A1
▸ Model answer. A ∩ B = {6}.
Question 2 · 2 marks · Non-calculator
|
“Using the same sets, list the members of A'.” |
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.
▸ Odd numbers seen. M1
▸ {1, 3, 5, 7, 9}. A1
▸ Model answer. A' = {1, 3, 5, 7, 9}.
Question 3 · 4 marks · Non-calculator
|
40 students were asked whether they take music (M) and art (R). 22 take music, 15 take art and 8 take both. Complete the Venn diagram. (4 marks) |
|
Question 3 · mark scheme
4 marks available. Award a mark for each point made.
▸ Overlap 8. B1
▸ M only 14. B1
▸ R only 7. B1
▸ Outside 11. B1
▸ Model answer. M only 14, overlap 8, R only 7, outside the circles 11.
Question 4 · 2 marks · Non-calculator
|
“One of the 40 students is chosen at random. Use your Venn diagram to work out the probability that the student takes neither music nor art.” |
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.
▸ 11. M1
▸ 11/40. A1
▸ Model answer. 11/40.
Question 5 · 2 marks · Non-calculator
|
“Write down the number of students in M ∪ R.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ Adding the three regions. M1
▸ 29. A1
▸ Model answer. 14 + 8 + 7 = 29.
Question 6 · 3 marks · Non-calculator
|
“One of the 40 students is chosen at random. Given that the student takes art, work out the probability that they also take music.” |
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.
▸ 15 as the total. M1
▸ 8/15. A1
▸ A correct conclusion. C1
▸ Model answer. 15 students take art, and 8 of these also take music. The probability is 8/15.