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4024/21 Oct/Nov 2020, Question 8: Conditional Probability

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

Bag has 36 balls, n red and rest green; two drawn without replacement. Complete tree diagram, form and solve a quadratic, then find probability both balls are green given more green than red.

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

Part (a) · 2 marks

Fill in the missing branch probabilities on the second-ball tree diagram, using n and 36.

  1. 1After a green first ball, 35 balls are left and n are red
  2. 2Red: n/35, Green: (35 − n)/35

Answern/35 and (35−n)/35 correctly positioned on tree diagram

Part (b) · 1 mark

Write an expression in n for the probability first ball red and second ball green.

  1. 1P(red then green) = n/36 × (36 − n)/35

Answern/36×(36−n)/35

Part (c) · 2 marks

Given that this probability equals 1/7, show it rearranges to n² − 36n + 180 = 0.

  1. 1n(36 − n)/(36 × 35) = 1/7
  2. 27n(36 − n) = 1260
  3. 336n − n² = 180
  4. 4n² − 36n + 180 = 0

AnswerCorrect rearrangement to n² − 36n + 180 = 0

Part (d) · 2 marks

Solve n² − 36n + 180 = 0, showing your working.

  1. 1Find two numbers that multiply to 180 and add to 36: 6 and 30.
  2. 2n² − 36n + 180 = (n − 6)(n − 30) = 0
  3. 3n − 6 = 0 or n − 30 = 0, so n = 6, 30

Answer6, 30

Part (e) · 3 marks

Using the condition that green balls outnumber red, find the probability both balls drawn are green as a lowest-terms fraction.

  1. 1More green than red, so n = 6 and there are 30 green balls
  2. 2P(green, green) = 30/36 × 29/35 = 870/1260
  3. 3= 29/42

Answer29/42

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