Venn diagrams and set notation.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Venn diagrams and set notation

Probability

Lesson 6 of 6

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Probability

Lesson 6 of 6

Answer each one, then check.

1

List the even numbers from 1 to 10.

2

List the multiples of 3 from 1 to 10.

3

What number is in both lists above?

4

Work out 40 − 29.

5

Simplify 8/16.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Probability

Lesson 6 of 6

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Probability

Lesson 6 of 6

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).

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Set Notation

Probability

Lesson 6 of 6

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Shading Set Regions

Probability

Lesson 6 of 6

Shade the region that matches the notation.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Sets of Numbers

Probability

Lesson 6 of 6

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Completing a Venn Diagram

Probability

Lesson 6 of 6

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Survey

Probability

Lesson 6 of 6

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probabilities from a Venn Diagram

Probability

Lesson 6 of 6

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

HIGHER TIER

Conditional Probability from a Venn Diagram

Restrict to the given set.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Higher

Given That

Probability

Lesson 6 of 6

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

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Probability

Lesson 6 of 6

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'.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probability

Lesson 6 of 6

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.

WHAT 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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Probability

Lesson 6 of 6

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.

openrevise.com

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

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.

openrevise.com