A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 3 is obtained. The number of throws taken is denoted by the random variable X.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] P(X=8)= (5/6)⁷ * (1/6) = 0.0465, 78125/1679616
2. [M1] 1-(5/6)ᵈ, d=8,9.
3. [A1] = 0.767 AWRT.
4. [M1] (1/6) + (5/6)(1/6) + ... + (5/6)⁷(1/6)
5. [A1] = 0.767 AWRT.
6. [B1] Two identified scenarios correct, accept un-simplified calculations.
7. [M1] Four correctly identified scenarios added (here they may be identified by correct un-simplified calculations). Condone ²C₁ = 2 etc.
8. [A1] 0.196, 763/3888 (0.196 < p < 0.1963. SCB1 for correct answer if M mark not awarded.)
9. [B1] 1 - 1 identified scenario correct, accept un-simplified calculations.
10. [M1] 1 - 2 correctly identified scenarios (here they may be identified by correct un-simplified calculations). Condone ⁵C₁ = 5.
11. [A1] 0.196, 763/3888 (0.196 < p < 0.1963. SCB1 for correct answer if M mark not awarded.)