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Probability & Venn Diagrams (Cambridge O Level Maths)

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Paul

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Probability & Venn Diagrams

What is a Venn diagram?

  • Venn diagrams allow us to show two (or more) characteristics of a situation where there is overlap between the characteristics
  • For example, students in a sixth form can study biology or chemistry but there may be students who study both

What might I be asked to do with a Venn diagram?

  • You can be asked to
    • draw a Venn diagram and/or
    • interpret a Venn diagram
  • Strictly speaking the rectangle (box) is always essential on a Venn diagram
    • it represents everything that can happen in the situation
    • you may see the letter calligraphic E or xi written inside or just outside the box
      • this means “the set of all possible outcomes” - i.e. “everything”!
      • sometimes the letters U or S are used instead
  • The words AND and OR become very important in both drawing and interpreting Venn diagrams
  • You will need to be familiar with the symbols ∩ and ∪
    • ∩ is intersection
    • ∪ is union
    • these mean AND and OR (respectively)

How do I draw a Venn diagram?

  • Start with a “box” and overlapping “bubbles”
    • the number of bubbles needed will depend on how many characteristics you are dealing with
    • it will usually be 2 or 3
  • Work through each sentence/piece of information given in a question to begin completing sections of your Venn diagram
    • pieces of information may have to be combined before you can enter a value into the diagram
    • not all values will be given directly
      • some may need working out
      • you will be expected to do this to complete your Venn diagram
  • Remember to consider AND and OR

How do I interpret a Venn diagram?

  • Use the information in the question to identify the parts of the Venn diagram needed to answer it
  • Shading the relevant parts of a Venn diagram can be helpful
  • Be careful with probability notation such as
    • A' (not A)
    • the symbols ∩ and ∪

How do I use Venn diagrams with conditional probability?

  • Probabilities that only involve a subset of the things in a Venn diagram are called conditional probabilities
    • For example with students studying German or Spanish or both, you might want to know the probability that a student who studies Spanish also studies German
    • This would require considering only the 'Spanish' part of the Venn diagram when finding the probability
  • Conditional probability questions are often (but not always!) introduced by the expression 'given that...'
    • For example 'Find the probability that a randomly chosen student studies German, given that the student also studies Spanish'
    • The answer would be the number in the 'Spanish AND German' part of the diagram divided by the sum of all the numbers in the 'Spanish' part of the diagram
  • Conditional probabilities are sometimes written using the 'straight bar' notation straight P open parentheses A vertical line B close parentheses 
    • That is read as 'the probability of A given B'
    • For example straight P open parentheses German vertical line Spanish close parentheses would be the probability that a student studies German, given that the student also studies Spanish
      8-1-6-cp-notes-fig3a-new

Examiner Tip

  • You may have to use your Venn diagram more than once in a question
    • so shading the original diagram can become confusing if you're trying to use it more than once
    • draw a 'mini'-Venn diagram (a small quick sketch just showing the box and bubbles but no values) and shade that

Worked example

In a class of 30 students, 15 students study Spanish, and 3 of the Spanish students also study German.
7 students study neither Spanish nor German.

a)

Draw a Venn diagram to show this information.

We start with the 3 in the intersection ("overlap"); we can then deduce the "Spanish only" section is 12.
7 needs to be outside both bubbles but within the box.
With a total of 30 we can work out how many students study "German only" and complete the diagram.

b)

Use your Venn diagram to find the probability that a student, selected at random from the class, studies Spanish but not German.

Highlight the part "Spanish only".

Venn-Q1b-Working, downloadable IGCSE & GCSE Maths revision notes

Pick out the numbers you need carefully.

Students studying "Spanish only" = 12
Total number of students = 30

P(Spanish only) bold equals bold 12 over bold 30 stretchy left parenthesis equals 2 over 5 stretchy right parenthesis 

Worked example

Given the Venn diagram below, find the following probabilities: Venn Q2, IGCSE & GCSE Maths revision notes

a)

P(A)

Draw a 'mini'-Venn diagram - a quick sketch without the details.
See these questions as "ways to win" - so in this part you win if in "bubble A".
B and C do not come into it at all.

venn-q2a-sketch-2

Total in A = 3 + 5 + 1 + 8 =17
Total = 3 + 5 + 1 + 8 + 2 + 4 + 8 + 9 = 40

Error converting from MathML to accessible text.

b)

P(A ∩ B ∩ C )

∩ - intersection - AND - "win" if "in A" AND "in B" AND "in C".

venn-q2b-sketch-2

Total in A AND B AND C = 8
Total = 40

c)

P(B' ∩ C )

∩ - intersection - AND - "win" if "not in B" AND "in C".

venn-q2c-sketch-2

Total in "not B" AND C = 5
Total = 40

Error converting from MathML to accessible text.

d)

P(A ∪ B )

∪ - UNION - OR - "win" if "in A" OR "in B".

venn-q2d-sketch-2

Total in A OR B = 3 + 1 + 8 + 5 + 2 + 8 = 27
Total = 40

Error converting from MathML to accessible text.

e)

P(A ∪ B ∪ C )

∪ - union - OR - "win" if "in A" OR "in B" OR "in C".

venn-q2e-sketch-2

In this case, it will be easier to subtract from the whole total.

Total in A OR B OR C = 40 - 9 = 31
Total = 40

f)

P(A' ∪ B' )

∪ - union - OR - "win" if "not in A" OR "not in B".
This one is particularly difficult to see without a diagram!

venn-q2f-sketch-2

Total in A' OR B '  = 40 - 8 - 5 = 27
Total = 40

Worked example

CP Example fig2 sol (1), downloadable IGCSE & GCSE Maths revision notes

CP Example fig2 sol (2), downloadable IGCSE & GCSE Maths revision notes

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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.