Seva has a coin which is biased so that when it is thrown the probability of obtaining a head is ¹⁄₃. He also has a bag containing 4 red marbles and 5 blue marbles. Seva throws the coin. If he obtains a head, he selects one marble from the bag at random. If he obtains a tail, he selects two marbles from the bag at random and without replacement.

Mark scheme

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

1. [B1] Two of the calculations for P(HR), P(TBR), either P(TR) or P(TRR) + P(TRB) unsimplified, ignore any identification. Condone 4/8=1/2 in the unsimplified calculation. Condone use of tree diagram to show calculation if values correct at end.
2. [M1] Values of all correct identified scenarios added. Correct branches may be identified on the tree diagram.
3. [A1] 0.6296..., 0.630 If M0 scored SC B1 for acceptable answers, WWW.
4. [(B1)] One calculation of P(HB), P(TBB), unsimplified, ignore any identification. 1 - probability must be seen. Condone use of tree diagram to show calculation if values correct at end.
5. [(M1)] 1-values of two correct identified scenarios subtracted. Correct branches may be identified on the tree diagram.
6. [(A1)] 0.6296..., 0.630 If M0 scored SC B1 for acceptable answers, WWW.
7. [(B1)] Calculation for P(T, (1 – no R)) seen unsimplified. Condone use of tree diagram to show calculation if values correct at end.
8. [(M1)] Values of two correct identified scenarios added. Correct branches may be identified on the tree diagram.
9. [(A1)] 0.6296..., 0.630 If M0 scored SC B1 for acceptable answers, WWW.
10. [M1] P(head ∩ no reds) / P(no reds) = (1/3)*(5/9) / (1-their (a)) or d/10/27 or d/27 or d/(1-17/27), 0<d<1. Condone 10/27 = 0.3704 or more accurate.
11. [A1] OE Condone 0.499[9...].
12. [(M1)] P(head ∩ no reds) / (P(HB) + P(TBB)) = (1/3)*(5/9) / [(1/3)*(5/9) + (2/3)*(5/9)*(4/8)] or d/10/27 or d/27 or d/(10/27), 0<d<1. Condone 10/27 = 0.3704 or more accurate.
13. [(A1)] OE Condone 0.499[9...].