Tim has two bags of marbles, A and B. Bag A contains 8 white, 4 red and 3 yellow marbles. Bag B contains 6 white, 7 red and 2 yellow marbles. Tim also has an ordinary fair 6-sided dice. He rolls the dice. If he obtains a 1 or 2, he chooses two marbles at random from bag A, without replacement. If he obtains a 3, 4, 5 or 6, he chooses two marbles at random from bag B, without replacement.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [M1] Either 2/6 × 8/15 × 7/14 or 4/6 × 6/15 × 5/14 seen, accept unsimplified.
2. [M1] q/6 × r/15 × (r-1)/14 + (6-q)/6 × s/15 × (s-1)/14 seen, no additional terms, accept unsimplified. Condone q/6 × r/15 × r/15 + (6-q)/6 × s/15 × s/15. 1 < q < 5, 1 < r, s < 9.
3. [A1] 58/315 or 0.184 SC B1 for 58/315 if either M mark withheld.
4. [B1] P(W & R from bag B) = 4/6 × 6/15 × 7/14 or 2 × 4/6 × 6/15 × 7/14 [= 1/15 or 0.267] Seen alone or as numerator/denominator of conditional probability.
5. [M1] P(WR or RW) = P(W & R from bag A) + P(W & R from bag B) = a × 2/6 × 8/15 × 4/14 + a × 4/6 × 6/15 × 7/14 + their P(W & R from bag B). a = 1 or 2. [expect 116/315 or 0.368] Seen alone or as numerator/denominator of conditional probability.
6. [M1] their identified P(W & R from bag B) / their identified P(W & R or RW) Accept unsimplified.
7. [A1] 0.7241379 to at least 3SF.