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## Venn diagrams

A Venn diagram is a graphical way of representing the relationships between sets. In each Venn diagram a set is represented by a closed curve. The region inside the curve represents the elements that belong to the set, while the region outside the curve represents the elements that are excluded from the set.

Venn diagrams are helpful for thinking about probability since we deal with different sets. Consider two events, A and B, in a sample space S. The diagram below shows the possible ways in which the event sets can overlap, represented using Venn diagrams:

The sets are represented using a rectangle for S and circles for each of A and B. In the first diagram the two events overlap partially. In the second diagram the two events do not overlap at all. In the third diagram one event is fully contained in the other. Note that events will always appear inside the sample space since the sample space contains all possible outcomes of the experiment.

Venn diagrams and probabilities

## Example 1: Venn diagrams

### Question

Represent the sample space of two rolled dice and the following two events using a Venn diagram:

• Event A: the sum of the dice equals 8

• Event B: at least one of the dice shows

## Example 2: Venn diagrams

### Question

Consider the set of diamonds removed from a deck of cards. A random card is selected from the set of diamonds.

• Write down the sample space, S, for the experiment.

• What is the value of $n(S)$?

• Consider the following two events:

• P: An even diamond is chosen

• R: A royal diamond is chosen

Represent sample space S and events P and R using a Venn diagram.

#### Write down the sample space S

$S={A;2;3;4;5;6;7;8;9;10;J;Q;K}$(1)

#### Write down the value of $n(S)$

$n(S)=13$(2)

## Exercise 1:

Let S denote the set of whole numbers from 1 to 16, X denote the set of even numbers from 1 to 16 and Y denote the set of prime numbers from 1 to 16. Draw a Venn diagram depicting S, X and Y.

There are 79 Grade 10 learners at school. All of these take some combination of Maths, Geography and History. The number who take Geography is 41, those who take History is 36, and 30 take Maths. The number who take Maths and History is 16; the number who take Geography and History is 6, and there are 8 who take Maths only and 16 who take History only.

1. Draw a Venn diagram to illustrate all this information.

2. How many learners take Maths and Geography but not History?

3. How many learners take Geography only?

4. How many learners take all three subjects?

Pieces of paper labelled with the numbers 1 to 12 are placed in a box and the box is shaken. One piece of paper is taken out and then replaced.

1. What is the sample space, S?

2. Write down the set A, representing the event of taking a piece of paper labelled with a factor of 12.

3. Write down the set B, representing the event of taking a piece of paper labelled with a prime number.

4. Represent A, B and S by means of a Venn diagram.

5. Find

1. $nS$

2. $nA$

3. $nB$

A

S, containing X and Y inside.

Only X: 4,6,8,10,12,14,16

Only Y: 3,5,7,11,13

X and Y: 2

In S but neither X nor Y: 1,9,15

B

n(S) = 16

n(X) = 8

n(Y) = 6

n(X U Y) = 1

n(X n Y) = 13

A - Three overlapping circles. Label them M, G and H respectively

B - Each student must do exactly one of the following:

• Take only geography;
• Only take maths and/or history;
There are 30 + 36 - 16 = 50 students doing the second one, therefore there must be 79 - 50 = 29 only doing geography.

Each students must do exactly one of:
• Only take geography;
• Only take maths;
• Take history;
• Take geography and maths, but not history;
There are 29, 8 and 36 of the first three. so the answer to B is:

79 -29 - 8 - 36 = 6 people

D: Each student must do exactly one of

• Do geography
• Only do maths
• Only do history
• Do maths and history but not geography
Using the same method as before, the number of people in the last group is:

79 - 41 - 8 - 16 = 14

But, 16 people do maths and history, so there must be 16 - 14 people who do all three.

A: S = {1,2,...,12}

B: A = {1,2,3,4,6,12}

C: B = {2,3,5,7,11}

D: obvious

E i)12

ii)6

iii)5

iv)2

v)9

F yes