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

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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 

      • 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 apostrophe is represented by the regions that are not in the A circle

  • A intersection B is represented by the region where the A and B circles overlap

  • A union 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
3-1-2-fig2-various-venns-part-2

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

    • straight P left parenthesis A intersection B right parenthesis equals straight P left parenthesis A right parenthesis straight P left parenthesis B right parenthesis

  • To find the conditional probability straight P left parenthesis A vertical line B right parenthesis

    • 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

Examiner Tips and Tricks

  • 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.

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

b) 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

c) 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: straight P left parenthesis A union B right parenthesis equals straight P left parenthesis A intersection B right parenthesis plus straight P left parenthesis A intersection B apostrophe right parenthesis plus straight P left parenthesis A apostrophe intersection B right parenthesis

    • subtract the probabilities of the combined outcomes that are not part of that event from 1

      • For example: straight P left parenthesis A union B right parenthesis equals 1 minus straight P left parenthesis A apostrophe intersection B apostrophe right parenthesis

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

Examiner Tips and Tricks

  • 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.

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

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

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

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

4-3-3-ib-ai-aa-sl-tree-diagram-c-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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