Discia GCSE Maths › Probability › Venn Diagrams

Venn Diagrams worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

10 questions 38 marks Sheet 1
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  1. 4 marks

    In a group of 22 students, 16 study French and 7 study German.

    2 of the students study neither language.

    You may use the Venn diagram to help you.

    An empty two-set Venn diagram inside a rectangle, the circles labelled French and GermanFrenchGermanξ
    (a)

    How many of the students study both languages?

    (b)

    One of the students is chosen at random.

    Find the probability that this person is one of those who study French but not German.

  2. 5 marks

    The Venn diagram shows information about a group of pupils.

    A completed two-set Venn diagram with 16 in Piano only, 4 in the overlap, 14 in Guitar only and 1 outside both circlesPianoGuitar16414ξ1
    (a)

    How many pupils are in the group altogether?

    (b)

    One of the pupils is chosen at random.

    Find the probability that this person is one of those who play the piano.

    (c)

    One of the pupils is chosen at random.

    Find the probability that this person is in exactly one of the two circles.

  3. 1 mark

    The shaded region of the Venn diagram shows a set.

    A three-set Venn diagram with one region shaded by hatchingABCξ

    Use set notation to write down the shaded set.

  4. 1 mark

    Shade the region \(A \cap B\) on the Venn diagram.

    An empty two-set Venn diagram inside the universal set rectangleABξ
  5. 5 marks

    48 members were asked which of the gym, the pool and the studio they use.

    The Venn diagram shows the results.

    A completed three-set Venn diagram for Gym, Pool and Studio, with 2 in the middle region and 4 outside all three circlesGymPoolStudio95103269ξ4
    (a)

    How many of the members use all three?

    (b)

    How many of the members use exactly one of the three?

    (c)

    One of the members is chosen at random.

    Work out the probability that they use the gym and the studio.

  6. 5 marks

    The universal set \(\xi\) is the 21 students in a club.

    The Venn diagram shows how many of them are in set \(A\) and how many are in set \(B\).

    A two-set Venn diagram for sets A and B inside the universal set, with 6 in A only, 4 in the overlap, 5 in B only and 6 outside bothAB645ξ6

    One of the 21 students is chosen at random.

    (a)

    Find \(P(A \cap B)\).

    (b)

    Find \(P(A \cup B)\).

    (c)

    Find \(P(A')\).

  7. 3 marks

    The Venn diagram shows information about a group of adults. Each number is how many adults are in that region.

    A two-set Venn diagram with 12 in Coffee only, x in the overlap, 8 in Tea only and 4 outside both circlesCoffeeTea12x8ξ4

    One of the adults is chosen at random.

    The probability that this adult is in both Coffee and Tea is \(\frac{1}{ 7 }\).

    Find the value of \(x\).

  8. 5 marks

    44 members were asked which of the gym, the pool and the studio they use.

    • 21 use the gym
    • 19 use the pool
    • 22 use the studio
    • 9 use both the gym and the pool
    • 8 use both the gym and the studio
    • 11 use both the pool and the studio
    • 4 use all three

    You may use the Venn diagram to help you.

    An empty three-set Venn diagram inside a rectangle, the circles labelled Gym, Pool and StudioGymPoolStudioξ
    (a)

    How many of the members use none of the three?

    (b)

    How many of the members use the gym only?

  9. 4 marks

    In a group of 34 pupils, 16 play the piano and 17 play the guitar.

    5 of the pupils play neither instrument.

    You may use the Venn diagram to help you.

    An empty two-set Venn diagram inside a rectangle, the circles labelled Piano and GuitarPianoGuitarξ
    (a)

    How many of the pupils play both instruments?

    (b)

    One of the pupils is chosen at random.

    Find the probability that this person is one of those who play the piano but not the guitar.

  10. 5 marks

    The Venn diagram shows information about a group of children.

    A completed two-set Venn diagram with 16 in Dog only, 4 in the overlap, 6 in Cat only and 7 outside both circlesDogCat1646ξ7
    (a)

    How many children are in the group altogether?

    (b)

    One of the children is chosen at random.

    Find the probability that this person is one of those who have a dog.

    (c)

    One of the children is chosen at random.

    Find the probability that this person is in exactly one of the two circles.

Answers

  1. (a) \(3\)
    (b) \(\frac{13}{22}\)
  2. (a) \(35\)
    (b) \(\frac{4}{7}\)
    (c) \(\frac{6}{7}\)
  3. \(B \cap C\)
  4. Shade only the overlap of the two circles.
  5. (a) \(2\)
    (b) \(28\)
    (c) \(\frac{5}{48}\)
  6. (a) \(\frac{4}{21}\)
    (b) \(\frac{5}{7}\)
    (c) \(\frac{11}{21}\)
  7. \(4\)
  8. (a) \(6\)
    (b) \(8\)
  9. (a) \(4\)
    (b) \(\frac{6}{17}\)
  10. (a) \(33\)
    (b) \(\frac{20}{33}\)
    (c) \(\frac{2}{3}\)

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