Venn Diagrams & Tree Diagrams (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

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 (or lack of) between possible outcomes

• A Venn diagram consists of

  • A rectangle representing the sample space

  • A circle for each event

    • Circles may or may not overlap depending on which outcomes are shared between events

How are Venn diagrams labelled and what do the numbers inside mean?

• The rectangle represents the sample space (all possible outcomes from the experiment)

  • It is often referred to as the Universal Set and is commonly labelled with  (the Greek lower case letter Xi) or (Kunstler script font)
    There is no standardised symbol used for this purpose

• Circles are labelled with their event name (A, B, etc.)

• The numbers inside a Venn diagram (there should be one in each region) will represent either a frequency or a probability

  • In the case of probabilities being shown, all values should total 1

What do the different regions and bubbles overlapping mean on a Venn diagram?

• This will depend on how many events there are and how the outcomes overlap

• Venn diagrams show ‘AND’ and ‘OR’ statements easily

• Venn diagrams also instantly show mutually exclusive events

  • Independence can be deduced from the probabilities involved

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

  • I.e. fill in intersections (overlaps) first

  • This is particularly crucial with Venn diagrams with three events

    • If all three circles overlap, the intersection of events A and B  will include the intersection of events A, B  and C

    • A question would make it clear if a given frequency or probability is only for events A and B , and not C

• Any frequencies or probabilities not given may be able to be calculated from those that are

  • Use the results from basic probability to deduce missing frequencies or probabilities and answer questions

• Check a completed Venn diagram that the frequencies sum to the total involved or that probabilities sum to 1

Tree Diagrams

What is a tree diagram?

• A tree diagram is used to

  • Show the (combined) outcomes of more than one event that happen one after the other

  • Help calculate probabilities when AND and/or ORs are involved

• Tree diagrams are mostly used when there are only two mutually exclusive outcomes of interest

  • E.g. “Rolling a 6 on a dice” and “Not rolling a 6 on a dice”

• More than three outcomes per event can be shown on a tree diagram but they soon become difficult to draw and so lose their effectiveness

• Tree diagrams are very helpful when probabilities for a second event change depending on the first event

How do I draw and label a tree diagram?

• In the second experiment, P(B)  may be different on the top set of branches than the bottom set

  • This is because the top set of branches follow on from event A but the bottom set of branches follow on from event "not A"

  • E.g. If a red ball is taken from a bag and not replaced in the first experiment, then the probability of picking a red ball in the second experiment has now changed

• Sometimes a second branch may not be needed following a first event

  • E.g. In aiming to pass a test, the event fail on the first attempt would require a second attempt but the event pass on the first attempt would not

How do I solve probability problems involving tree diagrams?

• Interpret questions in terms of AND and/or OR

• Draw, or complete a given, tree diagram

  • Determine any missing probabilities

    • Use

    • Consider if probabilities for the 2nd experiment change depending on the outcome from the 1^st experiment

• Write down the (final) outcome of the combined events and work out their probabilities

  • These are AND statements

  • 

    • Do not simplify fractions yet

• If more than one (final) outcome is required to answer a question then add their probabilities

  • These are OR statements

  •  

    • This applies since all the (final) outcomes are mutually exclusive

    • Note that  and are implied AND statements (for example AB means  A and B)