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4MA1/2H Jun 2022, Question 16: Three-Set Venn Diagrams

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2H (Higher) · Calculator · 8 marks

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The question, in short

Use survey data on 100 farmers keeping goats, sheep and chickens to fill a three-set Venn diagram, then find set sizes and a probability.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 3 marks

Complete the Venn diagram with the number of farmers in each subset using the given counts.

  1. 1All three: 6
  2. 2Two only: G and S 11 − 6 = 5, S and C 17 − 6 = 11, G and C 18 − 6 = 12
  3. 3S only: 31 − 5 − 6 − 11 = 9. C only: 53 − 11 − 6 − 12 = 24
  4. 4In at least one set: 100 − 20 = 80, so G only = 80 − (6+5+11+12+9+24) = 13
  5. 5G only 13, S only 9, C only 24, G and S only 5, S and C only 11, G and C only 12, all three 6, outside 20

AnswerG only: 13; S only: 9; C only: 24; G and S only: 5; S and C only: 11; G and C only: 12; all three: 6; outside all three: 20

Part (bi) · 1 mark

Find n(G), the number of farmers with goats.

  1. 1n(G) = G only + G∩S only + G∩C only + G∩S∩C
  2. 213 + 5 + 12 + 6 = 36

Answer36

Part (bii) · 1 mark

Find n([G ∪ S]′), the number outside goats or sheep.

  1. 1n(G) = 36, n(S) = 31, and G and S overlap in 5 + 6 = 11
  2. 2n(G ∪ S) = 36 + 31 − 11 = 56
  3. 3n([G ∪ S]′) = 100 − 56 = 44

Answer44

Part (biii) · 1 mark

Find n(G′ ∩ C), the number with chickens but no goats.

  1. 1n(Gʹ ∩ C) = those in C but not in G
  2. 2C only + S∩C only = 24 + 11 = 35

Answer35

Part (c) · 2 marks

Given a farmer with chickens is chosen at random, find the probability they also have goats.

  1. 1Farmers with chickens: 53
  2. 2Farmers with chickens and goats: G∩C only + G∩S∩C = 12 + 6 = 18
  3. 3Probability = 18 / 53

Answer18/53

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