Sample Space Diagrams (DP IB Maths: AA HL)

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Dan

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Maths

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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
- a circle for each event
  - Circles may or may not overlap depending on which outcomes are shared between events
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

What do the different regions mean on a Venn diagram?

$A'$ is represented by the regions that are not in the A circle
$A \cap B$ is represented by the region where the A and B circles overlap
$A \cup B$ 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

3-2-1-fig1-venn-and-set-notation

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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
If two events are independent you can use the formula
- $P (A \cap B) = P (A) P (B)$
To find the conditional probability $P (A | B)$
- 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

plx9zFR~_3-2-2-fig1-set-notation-examples

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.

Draw a Venn diagram to represent the information.

4-3-3-ib-ai-aa-sl-venn-diagram-a-we-solution

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

4-3-3-ib-ai-aa-sl-venn-diagram-b-we-solution

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

4-3-3-ib-ai-aa-sl-venn-diagram-c-we-solution

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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
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

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: $P (A \cup B) = P (A \cap B) + P (A \cap B') + P (A' \cap B)$
- subtract the probabilities of the combined outcomes that are not part of that event from 1
  - For example: $P (A \cup B) = 1 - P (A' \cap B')$

UclzomJM_3-2-3-fig1-tree-setup

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 A given B then put B on the first set of branches
- If you have the probability of B given A then put A on the first set of branches

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.

Draw a tree diagram to represent the information.

4-3-3-ib-ai-aa-sl-tree-diagram-a-we-solution