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4024/21 May/Jun 2020, Question 9: Tree Diagrams for Dependent Events

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 12 marks

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The question, in short

Bag has 10 tiles: 4 red, x white, rest blue. Two tiles drawn without replacement. Complete tree diagram, find probabilities, solve quadratic, and find probability first tile is blue.

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

Part (a) · 2 marks

Fill in the missing second-tile probabilities on the tree diagram after a blue tile is drawn first.

  1. 1After a blue tile, 9 tiles remain with 4 red and x white
  2. 2Red = 4/9, White = x/9

Answer4/9, x/9

Part (b) · 1 mark

Calculate the probability that both tiles drawn are red.

  1. 1P(red, red) = 4/10 × 3/9 = 12/90

Answer12/90

Part (ci) · 4 marks

Show that the probability both tiles are the same colour equals (x² − 6x + 21)/45.

  1. 1P(R, R) = 4/10 × 3/9 = 12/90
  2. 2P(W, W) = x/10 × (x − 1)/9 = (x² − x)/90
  3. 3P(B, B) = (6 − x)/10 × (5 − x)/9 = (30 − 11x + x²)/90
  4. 4Total = (12 + x² − x + 30 − 11x + x²)/90 = (2x² − 12x + 42)/90
  5. 5= (x² − 6x + 21)/45

Answer(12+ x²− x +30−11x + x²)/90 leading to (x²−6x + 21)/45

Part (cii) · 1 mark

Given the same-colour probability is 16/45, show that x² − 6x + 5 = 0.

  1. 1(x² − 6x + 21)/45 = 16/45
  2. 2x² − 6x + 21 = 16
  3. 3x² − 6x + 5 = 0

Answer(x² − 6x + 21)/45 = 16/45 leading to x² − 6x + 5 = 0 with no errors

Part (ciii) · 2 marks

Solve the quadratic equation x² − 6x + 5 = 0.

  1. 1(x − 5)(x − 1) = 0
  2. 2x = 5 and x = 1

Answer5 and 1

Part (civ) · 2 marks

Using x = 1 and x = 5, with more red than white tiles, find the probability the first tile is blue.

  1. 1From part (c), x² − 6x + 5 = 0, so (x − 1)(x − 5) = 0
  2. 2x = 1 or x = 5; more red (4) than white, so x = 1
  3. 3Blue tiles = 10 − 4 − 1 = 5
  4. 4P(first tile blue) = 5/10

Answer5/10

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Also searched as: 4024/21/M/J/20 · 4024_s20_qp_21 · 4024 s20 21 · question 9.
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