Further Venn Diagrams (OCR A Level Maths: Statistics)

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Paul

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Paul

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Further Venn Diagrams

What do you mean by further Venn diagrams?

  • The Venn diagrams used here are no more complicated than those in the first Venn Diagrams revision note, however
    • Venn diagrams, and the wording/terminology in questions, may now involve the use of set notation including the symbols begin mathsize 16px style union end style(union),
       begin mathsize 16px style intersection end style(intersection), ‘ (complement) and | (“given that”)
    • Conditional probability problems can be solved using Venn diagrams

How do I solve conditional probability problems using Venn diagrams?

  • Interpreting questions in terms of AND (union), OR (intersection), complement ( ‘ ) and
    “given that” ( | )
  • Conditional probability may now be involved too
  • Use mini-Venn diagrams to sketch and shade the regions you are dealing with – use different colours if available or different styles of shading if not
    • Shading can help you see the answer

since straight P left parenthesis A vertical line B right parenthesis equals fraction numerator straight P left parenthesis A italic intersection B right parenthesis over denominator straight P left parenthesis B right parenthesis end fraction shade B first, then shade A intersection B

the answer will then be fraction numerator " double space shading " over denominator " all space shading " end fraction

3-2-2-fig1-set-notation-examples

Worked example

Three events, A comma space B and C are such that

        events A and C are mutually exclusive

        straight P left parenthesis A intersection B right parenthesis equals 0.2
straight P left parenthesis B intersection C right parenthesis equals 0.3
straight P left parenthesis left parenthesis A union B union C right parenthesis apostrophe right parenthesis equals 0.1
straight P left parenthesis B right parenthesis equals 0.7
straight P left parenthesis A apostrophe right parenthesis equals 0.75.

(a)
Draw a complete Venn diagram to show the probabilities connecting the three events.

 

(b)
Find 
(i)
straight P left parenthesis A vertical line B right parenthesis
(ii)
straight P left parenthesis A apostrophe vertical line C apostrophe right parenthesis
(iii)
straight P left parenthesis C vertical line left parenthesis A union B right parenthesis apostrophe right parenthesis
(a)
Draw a complete Venn diagram to show the probabilities connecting the three events.

3-2-2-fig2-we-solution-part-1

(b)
Find 
(i)
straight P left parenthesis A vertical line B right parenthesis
(ii)
straight P left parenthesis A apostrophe vertical line C apostrophe right parenthesis
(iii)
straight P left parenthesis C vertical line left parenthesis A union B right parenthesis apostrophe right parenthesis

3-2-2-fig2-we-solution-part-2-luggage-removed

Examiner Tip

  • Always draw the box in a Venn diagram; it represents all possible outcomes of the experiment so is a crucial part of the diagram, the bubbles merely represent the events we are particularly interested in
  • You may be able to answer some questions by applying formulae or you may prefer to use shaded mini-Venn diagrams; complicated questions tend to be easier with mini-Venn diagrams

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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.