### Venn Diagrams

**What is a Venn diagram?**

- A Venn diagram is a way to illustrate
**events**from an**experiment**and are particularly useful when there is an overlap between possible**outcomes** - A Venn diagram consists of
- a
**rectangle**representing the**sample****space (***U)*- The rectangle is labelled
*U* - Some mathematicians instead use
*S*or*ξ*

- The rectangle is labelled
- a
**circle**for each**event**- Circles may or may not overlap depending on which
**outcomes**are shared between**events**

- Circles may or may not overlap depending on which

- a
- The numbers in the circles represent either the
**frequency**of that event or the**probability**of that event- If the
**frequencies**are used then they should**add up to the total frequency** - If the
**probabilities**are used then they should**add up to 1**

- If the

**What do the different regions mean on a Venn diagram?**

- is represented by the regions that are
**not in**the*A*circle - is represented by the region where the
*A*and*B*circles**overlap** - is represented by the regions that
**are in***A*or*B*or both - Venn diagrams show ‘
**AND’**and ‘**OR’**statements easily - Venn diagrams also instantly show
**mutually****exclusive**events as these circles will**not overlap** **Independent**events can not be instantly seen- You need to use probabilities to deduce if two events are independent

**How do I solve probability problems involving Venn diagrams?**

- Draw, or add to a given Venn diagram, filling in as many values as possible from the information provided in the question
- It is usually helpful to work from the centre outwards
- Fill in
**intersections**(overlaps) first

- Fill in
- If two events are independent you can use the formula
- To find the conditional probability
- Add together the frequencies/probabilities in the
*B*circle- This is your denominator

- Out of those frequencies/probabilities add together the ones that are also in the
*A*circle- This is your numerator

- Evaluate the fraction

- Add together the frequencies/probabilities in the

#### Exam Tip

- If you struggle to fill in a Venn diagram in an exam:
- Label the missing parts using algebra
- Form equations using known facts such as:
- the sum of the probabilities should be 1
- P(A∩B)=P(A)P(B) if A and B are independent events

#### Worked Example

40 people are asked if they have sugar and/or milk in their coffee. 21 people have sugar, 25 people have milk and 7 people have neither.

a)

Draw a Venn diagram to represent the information.

b)

One of the 40 people are randomly selected, find the probability that they have sugar but not milk with their coffee.

c)

Given that a person who has sugar is selected at random, find the probability that they have milk with their coffee.

### Tree Diagrams

**What is a tree diagram?**

- A
**tree****diagram**is another way to show the outcomes of combined events- They are very useful for intersections of events

- The events on the branches must be
**mutually exclusive**- Usually they are an event and its complement

- The probabilities on the second sets of branches
**can depend**on the outcome of the first event- These are
**conditional probabilities**

- These are
- When selecting the items from a bag:
- The second set of branches will be the
**same**as the first if the items**are replaced** - The second set of branches will be the
**different**to the first if the items**are not replaced**

- The second set of branches will be the

**How are probabilities calculated using a tree diagram?**

- To find the probability that two events happen together you
**multiply**the corresponding probabilities on their branches- It is helpful to find the probability of all combined outcomes once you have drawn the tree

- To find the probability of an event you can:
**add together**the probabilities of the**combined outcomes**that are part of that event- For example:

**subtract**the probabilities of the combined outcomes that are not part of that event from 1- For example:

**Do I have to use a tree diagram?**

- If there are
**multiple events**or trials then a tree diagram can get big - You can break down the problem by using the words
**AND/OR/NOT**to help you find probabilities without a tree - You can speed up the process by only drawing parts of the tree that you are interested in

**Which events do I put on the first branch?**

- If the events
*A*and*B*are**independent**then the**order does not matter** - If the events
*A*and*B*are**not independent**then the**order does matter**- If you have the probability of
then put*A*given*B*of branches*B*on the first set - If you have the probability of
then put*B*given*A*of branches*A*on the first set

- If you have the probability of

#### Exam Tip

- In an exam do not waste time drawing a full tree diagram for scenarios with lots of events unless the question asks you to
- Only draw the parts that you are interested in

#### Worked Example

20% of people in a company wear glasses. 40% of people in the company who wear glasses are right-handed. 50% of people in the company who don’t wear glasses are right-handed.

a)

Draw a tree diagram to represent the information.

b)

One of the people in the company are randomly selected, find the probability that they are right-handed.

c)

Given that a person who is right-handed is selected at random, find the probability that they wear glasses.