Sample Space Diagrams (DP IB Applications & Interpretation (AI)): 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 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? 

• 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

• 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

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: 

  • 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 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