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4024/22 Oct/Nov 2022, Question 4: Tree Diagrams for Dependent Events

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 10 marks

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The question, in short

Work with sets: list union elements, count a complement-intersection, describe a shaded Venn region, complete a three-language Venn diagram for 25 students, and find two selection probabilities.

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

Part (ai) · 1 mark

List the integers from 10 to 40 that are multiples of 6 or square numbers.

  1. 1P = {12, 18, 24, 30, 36}, Q = {16, 25, 36}
  2. 2P ∪ Q = {12, 16, 18, 24, 25, 30, 36}

Answer12, 16, 18, 24, 25, 30, 36

Part (aii) · 1 mark

Find how many integers from 10 to 40 are square numbers but not multiples of 6.

  1. 1P′ ∩ Q = square numbers that are not multiples of 6 = {16, 25}
  2. 2n(P′ ∩ Q) = 2

Answer2

Part (b) · 1 mark

Express the shaded region, all of C and D outside B, using set notation.

  1. 1Shaded: inside C or D, but outside B
  2. 2(D ∪ C) ∩ B′

Answer(D ∪ C) ∩ B′

Part (ci) · 2 marks

Complete the Venn diagram using 25 students, 7 French only, 8 Arabic, and all Arabic-Spanish students also study French.

  1. 1F only = 7
  2. 2A total 8: A only = 8 − 4 − 1 − 0 = 3 (S ∩ A only = 0, since all who study A and S also study F)
  3. 3F ∩ S only = 25 − (7 + 5 + 1 + 4 + 3 + 0 + 2) = 3
  4. 4F only 7, F ∩ S only 3, S ∩ A only 0, A only 3

AnswerF only: 7 F ∩ S only: 3 S ∩ A only: 0 A only: 3 (given: S only 5, F ∩ S ∩ A 1, F ∩ A only 4, outside 2)

Part (cii) · 2 marks

Find the probability that two students chosen at random both study Spanish only.

  1. 1P(both Spanish only) = 5/25 × 4/24
  2. 2= 20/600 = 1/30

Answer1/30

Part (ciii) · 3 marks

From students studying French, find the probability that exactly one of three chosen at random also studies Arabic.

  1. 1n(F) = 7 + 3 + 1 + 4 = 15; of these 5 study Arabic, 10 do not
  2. 2P(A, not, not) = 5/15 × 10/14 × 9/13 = 450/2730
  3. 3The Arabic student can be 1st, 2nd or 3rd: 3 × 450/2730 = 1350/2730
  4. 4= 45/91

Answer45/91

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Also searched as: 4024/22/O/N/22 · 4024_w22_qp_22 · 4024 w22 22 · question 4.
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