A-LevelMathematicsProbability and statistics9709/53 · Oct/Nov 2022 · Q78 Marks

Sam and Tom are playing a game which involves a bag containing 5 white discs and 3 red discs. They take turns to remove one disc from the bag at random. Discs that are removed are not replaced into the bag. The game ends as soon as one player has removed two red discs from the bag. That player wins the game. Sam removes the first disc.

(a)Find the probability that Tom removes a red disc on his first turn.[2]
(b)Find the probability that Tom wins the game on his second turn.[4]
(c)Find the probability that Sam removes a red disc on his first turn given that Tom wins the game on his second turn.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[M1] (3/8) * (2/7) + (5/8) * (3/7)
[A1] for 3/8 SC B1 for 3/8 with no explanation.
[M1] m/8 * n/7 * o/6 * q/5, 1 < m,n,o,q <= 5, m ≠ n ≠ o ≠ q
[A1] Probability for one scenario correct, accept unsimplified.
[M1] Adding probabilities for 3 correct scenarios and no incorrect.
[A1] Or 0.1071428... to 4SF or better. SC B1 for 3/28 with inadequate explanation.
[M1] their P(RRWR) / their 7(b) – must be a prob or 3/28
[A1]

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