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ASMaths: Probability & Statistics 1CIERevision Notes2. Probability2.1 Basic Probability2.1.2 Venn Diagrams

Venn Diagrams (CIE AS Maths: Probability & Statistics 1)

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Paul

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Venn Diagrams

What is a Venn diagram?

  • A Venn diagram is a way to illustrate events and are particularly useful when outcomes overlap
  • Venn diagrams are mostly used for 2 or 3 events
  • A Venn diagram consists of a box (rectangle) and a bubble (circle/ellipse) for each event
    • Bubbles may or may not overlap
    • Bubble(s) is not a technical term, but we like it!
  • The box represents all outcomes
    • It is often referred to as the Universal Set and is commonly labelled with the symbols calligraphic E (Kunstler script font uppercase letter E)size 16px S size 16px comma size 16px space size 16px U size 16px space size 16px or size 16px space size 16px xi  (Greek lowercase letter Xi)
    • There is no standardised symbol for this purpose
  • Bubbles 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

2-1-2-cie-fig1-basic-venns

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

2-1-2-cie-fig2-various-venns

Worked example

40 people were surveyed regarding which games consoles they owned.

8 people said they owned a Playstation 5 (P) and an Xbox Series X (X).

5 people said they owned neither of these consoles.

Of those people that owned only one games console, twice as many owned an Xbox Series X as a Playstation 5.

One of the 40 people is chosen at random.  Find the probability that this person 

(i)
owns both consoles,
(ii)
owns exactly one console,
(iii)
doesn’t own a Playstation 5.

2-1-2-cie-fig3-we-solution

Examiner Tip

  • The rectangle in a Venn diagram is a key part of the diagram
    • it represents all possible outcomes of the experiment
    • the bubbles merely represent the events we are particularly interested in
    • there is usually a few possibilities that fall outside of these events so this would be the section outside the bubbles but inside the rectangle
  • A quick ‘mini-Venn’ diagram shading the parts required to answer the question can be useful rather than always drawing a full Venn diagram with all its values

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.