A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2024 · Q46 Marks

Rahul has two bags, X and Y. Bag X contains 4 red marbles and 2 blue marbles. Bag Y contains 3 red marbles and 4 blue marbles. Rahul also has a coin which is biased so that the probability of obtaining a head when it is thrown is 1/4. Rahul throws the coin. • If he obtains a head, he chooses at random a marble from bag X. He notes the colour and replaces the marble in bag X. He then chooses at random a second marble from bag X. • If he obtains a tail, he chooses at random a marble from bag Y. He notes the colour and discards the marble. He then chooses at random a second marble from bag Y.

(a)Find the probability that the two marbles that Rahul chooses are the same colour.[3]
(b)Find the probability that the two marbles that Rahul chooses are both from bag Y given that both marbles are blue.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[B1] 2 clearly identified unsimplified probabilities from P(HRR), P(TRR), P(HBB) and P(TBB) correct.
[M1] Sum of 4 correct scenarios, may be identified by the unsimplified probability calculations.
[A1] (0.460317... to at least 3SF).
[M1] 3/14 / (3/14 + 1/36) oe,, 0.2142857... seen as numerator of a fraction, accept unsimplified, FT their P(TBB) from 4(a)
[M1] their 1/36 + their 3/14 FT from 4(a) or correct, 0.24206..., seen as denominator of a fraction, accept unsimplified.
[A1] Accept 0.8852589... rounded to at least 3SF. If one or both Ms not awarded, SC B1 for correct final answer WWW.

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