Recognize Ann rolls green on her first roll, using her die with 3 green faces out of 8 (M1)

eg \( \text{P(G)} = \frac{\text{number of green faces}}{\text{total faces}} \)

\( \frac{3}{8} \) (A1) (N2)

[2 marks]

(i) For Ann to win on her third roll, the sequence is: Ann red (\( \frac{5}{8} \)), Bob red (\( \frac{4}{8} \)), Ann red (\( \frac{5}{8} \)), Bob red (\( \frac{4}{8} \)), Ann green (\( \frac{3}{8} \)). Thus, \( p = \frac{5}{8}, q = \frac{4}{8} \) or \( q = \frac{4}{8}, p = \frac{5}{8} \) (A1A1) (N2)

(ii) Recognize Ann wins on her tenth roll after 9 losses (Ann red, Bob red) (M1)

eg Sequence: \( \left( \frac{5}{8} \times \frac{4}{8} \right)^9 \times \frac{3}{8} \)

\( k = 9 \) (number of loss pairs before Ann’s tenth roll) (A1) (N2)

Correct working to express probability: \( \frac{3}{8} \left( \frac{5}{8} \times \frac{4}{8} \right)^9 = \frac{3}{8} \left( \frac{20}{64} \right)^9 \) (A1)

\( r = \frac{20}{64} \left( = \frac{5}{16} \right) \) (A1) (N2)

[6 marks]

Recognize the probability is an infinite sum of Ann winning on her 1st, 2nd, 3rd, … roll (M1)

eg Sum of probabilities: Ann wins on \( 1^{\text{st}} \) roll or \( 2^{\text{nd}} \) roll or \( 3^{\text{rd}} \) roll…, \( S_\infty \)

Recognize geometric progression with first term and common ratio (M1)

First term: \( u_1 = \frac{3}{8} \) (Ann wins on first roll) (A1)

Common ratio: \( r = \frac{5}{8} \times \frac{4}{8} = \frac{20}{64} = \frac{5}{16} \) (probability both lose per round) (A1)

Correct substitution into infinite geometric series sum: \( \frac{u_1}{1 – r} = \frac{\frac{3}{8}}{1 – \frac{5}{16}} \) (A1)

Correct working: \( \frac{\frac{3}{8}}{\frac{11}{16}} = \frac{3}{8} \times \frac{16}{11} \) (A1)

\( \text{P (Ann wins)} = \frac{48}{88} \left( = \frac{6}{11} \right) \) (A1) (N1)

[7 marks]