S1: Venn DiagramsA Venn diagram is a way to represent sets of people or objects.

A Venn diagram consists of a rectangle (to represent the wholeset) with circles in it (to represent more interesting sets).

Example: Out of a group of 1000 students, 58 got A grades in A-Level Maths and 138 got B grades. Draw a

Venn diagram to show this.

represents whole setof 1000 people

represents Agrade students

represents Bgrade students

A58

B138

Write the remainingpeople in the top right of the main rectangle.

804

Venn diagrams are useful when it is possible to be in more thanone of the smaller sets, i.e. the sets are not mutually exclusive.

To show this, you need to make the circles that represent the smaller sets overlap – but the maths of the diagram changes.

Example: A restaurant holds 100 people. Ten have fish and chips. Thirty-three have chips and fifty-two have

fish. Draw a Venn diagram of this.

Fish Chips

How do we work outthe right numbers to put in the circles?

Start with the overlap.It was 10:

10

52 people are havingfish, but there arealready 10 people inthat circle. 42 remainin the ‘main’ part:4223 more for chips:

23

25

25 left:

S1: Venn Diagrams

You may be asked to draw a Venn diagram involving three sets.

Example: 3000 people are at a concert. 485 wear glasses (G), 2221 are female (F) and 241 have red hair (R). There are 14 female red-haired glasses wearers. 300 glasses wearers are women, 82 glasses wearers are red-haired. There are 76 red-haired females. Draw a Venn diagram to represent this.

Start with the sets.They must alloverlap:

R

G F

As before, now theoverlap figure (14):

14Fill in the gaps. 76red-haired females;14 already gone, so62:

62

Similarly, 68 in R and Gonly section:

68

97 left in R only:97286 in F and G only:

286

117 in G only:

1171859 in F only:

1859495 left over:

495

S1: Venn Diagrams

Questions are written using the proper notation. Remember:

“A ∩ B” means “both A and B happening at the same time” (said “A intersect B”). This is shown on a Venn diagram by the overlap of the circles.

A B

S1: Venn Diagrams

“A U B” means “either A or B” (said “A union B”). This is shown on a Venn diagram by everything inside the circles.

A B

S1: Venn Diagrams

“A´ ” means “everything that is not A” (said “A complement”). This is represented on a Venn diagram by everything outside the circle for A.

A B

S1: Venn Diagrams

Example: 30 students are asked if they have seen two films: Avatar (A) and X-Men: First Class (X).A ∩ X = 5. X = 17. 3 people have not seen either film.Draw a Venn diagram to represent this situation.

Start with the sets.They must overlap:

A XAs before, now theoverlap figure (5):

5Which gap can befilled in next? Xtotals 17. It has 5in it already, so 12must be left:

12

3 people have notseen either film:

3

The diagram musthave 30 people in it.There are 20 so far, so the A only sectionmust contain 10:10

Done! Now the tablecan be used toanswer questions.

What is the probability that a student chosen at random has seen Avatar?

S1: Venn Diagrams

Once complete, Venn diagrams can be used to find theprobability of certain events taking place.

Let’s look at the diagrams from this presentation so far.

A58

B138

804 What is the probability that a randomlychosen student has an A?

There are 58 A students out of 1000, so:

50029

100058

Fish Chips42

10

23

25What is the probability that a randomlychosen person is eating fish?

How many people are in the fish set?

42 + 10 = 52. There are 100 peoplein total, so:

2513

10052

S1: Venn Diagrams

Now for the last two diagrams:

R

G F14

6268

97

286117 1859

495 What is the probability a person chosenat random has either red hair or wearsglasses but not both?

Add the numbers in the R circle thataren’t in the overlap with G, and thenumbers in the G circle that aren’t inthe overlap with R:

97 + 62 + 117 + 286 = 562. 3000people in total, so:

1500281

3000562

A X512

3

10

What is the probability that a studentchosen at random has seen Avatar?

15 total in the A circle. 30 overall, so:

21

3015

S1: Venn Diagrams