O-LevelMathematicsProbability4024/12 · May/June 2017 · Q246 Marks
A bag contains n balls. 3 of the balls are white. Two balls are taken from the bag, at random, without replacement. [Figure: Incomplete tree diagram for drawing two balls without replacement.]
- (a)Complete the tree diagram. [Figure: Tree diagram with branches for First ball (White: 3/n, Not white: (n-3)/n), and Second ball (if First White: White: 2/(n-1), Not white: (n-3)/(n-1); if First Not white: White: 3/(n-1), Not white: (n-4)/(n-1)).][2]
- (b)The probability that both balls are white is 1/15. Show that n² − n − 90 = 0.[2]
- (c)Find the value of n.[2]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- [2] (a) Correctly completed tree diagram
- (n-3)/(n-1) oe
- (n-3)/n oe
- (n-4)/(n-1) oe
- C1 for one correct probability correctly positioned
- M1 (b) 3/n × 2/(n-1) = 1/15
- A1 Correct rearrangement with at least one further step to reach n²-n-90 = 0
- [2] (c) 10
- B1 for solutions 10, -9 seen
- or M1 for (n – 10)(n + 9) [ = 0]
- or for (1 ± √((-1)²-4 × 1 × -90))/(2 × 1) or better
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