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4024/22 May/Jun 2021, Question 11: Tree Diagrams for Dependent Events

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 9 marks

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The question, in short

Bag has 12 balls, x black and rest white; two drawn without replacement. Complete a tree diagram, find probability of one of each colour as a single fraction, then use both-black probability 14/33 to find x.

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

Part (a) · 2 marks

Fill in the missing probabilities on the two-stage tree diagram for drawing two balls without replacement.

  1. 1After a black ball: 11 left, 12 − x of them white → (12 − x)/11
  2. 2After a white ball: 11 left, x black → x/11
  3. 3and 11 − x white → (11 − x)/11

Answer(12 − x)/11, x/11, (11 − x)/11 correctly placed

Part (b) · 3 marks

Find an expression, as one fraction in x, for the probability of drawing one black and one white ball.

  1. 1P(BW) + P(WB) = x/12 × (12 − x)/11 + (12 − x)/12 × x/11
  2. 2= 2x(12 − x)/132
  3. 3= x(12 − x)/66

Answerx(12 − x)/66

Part (c) · 4 marks

Given both balls black has probability 14/33, form an equation in x, solve it, and state the number of black balls.

  1. 1x/12 × (x − 1)/11 = 14/33
  2. 2x(x − 1)/132 = 14/33, so x(x − 1) = 56
  3. 3x² − x − 56 = 0
  4. 4(x − 8)(x + 7) = 0
  5. 5x = 8 (x = −7 is impossible)

Answer8

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Also searched as: 4024/22/M/J/21 · 4024_s21_qp_22 · 4024 s21 22 · question 11.
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