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      Constructing three event Venn diagrams

      Lesson details

      Learning outcome

      I can construct a three event Venn diagram.

      Key learning points

      1. A three event Venn diagram is easier to complete if information from intersecting events is completed first
      2. Venn diagrams may show probabilities, frequencies or outcomes
      3. A three event Venn diagram may have some (or all) events being mutually exclusive
      4. A three event Venn diagram may have events which are subsets of other events

      Keywords

      • Outcome - An outcome is a result of a trial (e.g. getting heads when flipping a coin once or getting two heads when flipping a coin twice).

      • Sample space - A sample space is all the possible outcomes of a trial. A sample space diagram is a systematic way of producing a sample space.

      • Event - An event is a subset of a sample space. i.e. An outcome or set of outcomes that may occur from a trial (e.g. flipping a coin twice and getting the same result each time).

      • Mutually exclusive - Two or more events are mutually exclusive if they share no common outcome.

      • Venn diagram - Venn diagrams are a representation used to model statistical/probability questions. Commonly circles are used to represent events.

      Common misconception

      There is often a lot of information to process, in order to complete a three event Venn diagram accurately. Pupils may confuse which region to complete.

      A three event Venn diagram is easier to complete if information from intersecting events is completed first.

      Teacher tip

      Encourage pupils to draw/copy the Venn diagram and highlight or shade the regions that are needed for a given question.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      This probability Venn diagram shows the probability of a random spinner (spun once) landing on a square number or a factor of 36. Which of these probability statements are correct?

      An image in a quiz
      $$a + b + c = 1$$
      Correct answer: P(factor of 36) = $$b+c$$
      P(square ∪ factor of 36) = $$a+c$$
      Correct answer: P(factor of 36 ∪ square) = $$ 1-d$$
      P(factor of 36 |square) = $$ c$$

      Q2.
      This probability Venn diagram shows the probability of a random spinner (spun once) landing on a square number or a factor of 36. The value of $$x$$ is . Give your answer as a decimal.

      An image in a quiz
      Correct Answer: 0.06

      Q3.
      Here is a frequency Venn diagram. P(A ∩ B) = 0.2. The value of $$x$$ is .

      An image in a quiz
      Correct Answer: 3

      Q4.
      P(C given D) = 0.4. Which of these equations is correct to find the value of $$x$$?

      An image in a quiz
      Correct answer: $$\frac{2x}{4x+1}=0.4$$
      $$\frac{2x}{2x+1}=0.4$$
      $$\frac{2x+1}{4x+1}=0.4$$
      $$\frac{2x+1}{2x}=0.4$$

      Q5.
      A pack of cards contains 50 “character” (c) cards and some “spell” (s) cards. A card is taken at random, replaced and a 2nd card is taken. P(two spell cards) = 0.25. There are cards in the deck.

      An image in a quiz
      Correct Answer: 100

      Q6.
      The ratio of cats (c) to dogs (d) in an animal shelter is 4 : 3. Two random animals are taken from the shelter for adoption. Find an expression for P(2 dogs).

      An image in a quiz
      $$\frac{3}{7}$$
      $$\frac{9}{49}$$
      $$\frac{(3x-1)}{(7x-1)}$$
      Correct answer: $$\frac{3(3x-1)}{7(7x-1)}$$
      $$\frac{3(3x-1)}{49x}$$

      6 Questions

      Q1.
      A __________ space is all the possible outcomes of a trial.

      frequency
      population
      probability
      Correct answer: sample

      Q2.
      Izzy asks 30 pupils if they take GCSE Art (A), GCSE Drama (D) or GCSE Music (M). The frequencies are shown in the Venn diagram. pupils take music but not drama.

      An image in a quiz
      Correct Answer: 6, six

      Q3.
      Izzy asks 30 pupils if they take GCSE Art (A), GCSE Drama (D) or GCSE Music (M). The frequencies are shown in the Venn diagram. pupils do not take drama.

      An image in a quiz
      Correct Answer: 17, seventeen

      Q4.
      A spinner is numbered with the integers from 80 to 89. The incomplete outcome Venn diagram is used to display 3 events. Select the correct statements.

      An image in a quiz
      Event A is {multiples of 4}
      Correct answer: {prime numbers} is a subset of {odd numbers}
      {prime numbers} and {odd numbers} are mutually exclusive
      Correct answer: {prime numbers} and {multiples of 4} are mutually exclusive
      Correct answer: Event C has 3 elements.

      Q5.
      What region is shown shaded in this Venn diagram?

      An image in a quiz
      A'
      B' ∪ C'
      A ∪ B' ∪ C'
      Correct answer: A ⋂ B′ ⋂ C′
      A ∪ B′ ⋂ C′

      Q6.
      What region is shown shaded in this Venn diagram?

      An image in a quiz
      B ⋂ C′
      A ∪ (B ⋂ C)′
      Correct answer: (B ⋂ C)′
      A ∪ B′ ⋂ C′
      A ∪ B′ ∪ C

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