In a game, players attempt to score a goal by kicking a ball into a net. The probability that Leno scores a goal is 0.4 on any attempt, independently of all other attempts. The random variable X denotes the number of attempts that it takes Leno to score a goal.

Mark scheme

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

1. [B1]
2. [M1] (1-p)⁷ - (1-p)² or p² – p⁷ seen, 0 < p < 1.
3. [A1] If M0 awarded SC B1 0.3320064 or 25938/78125 CAO.
4. [M1] (1-p)xp²+(1-p)xp³ +(1-p)xp⁴+(1-p)xp⁵ +(1-p)xp⁶ seen, 0 < p < 1.
5. [A1] If M0 awarded SC B1 0.3320064 or 25938/78125 CAO.
6. [M1] 0.144(1-0.6⁵)/(1-0.6) or 0.144(1-p⁵)/(1-p) seen 0 < p < 1.
7. [A1] If M0 awarded SC B1 0.3320064 or 25938/78125 CAO.
8. [M1] 2 correct unsimplified outcomes. Condone not identified but not incorrectly identified.
9. [M1] Add values for 4 identified correct scenarios. Condone adding values of 2nd, 3rd and 4th attempts only. No incorrect scenarios.
10. [A1] If either or both M marks not awarded, SC B1 for 0.663, 2072/3125 WWW condone 1 index error.
11. [M1] At least 2 correct unsimplified terms.
12. [M1] Add values for 4 terms of the form ⁵Cₐ(0.4)ᵃ(0.6)⁵⁻ᵃ or 1 – sum of 2 terms of the form ⁵Cₐ(0.4)ᵃ(0.6)⁵⁻ᵃ.
13. [A1] If either or both M marks not awarded, SC B1 for 0.663 www condone 1 index error.
14. [B1] 30 and 18 seen, allow unsimplified. May be seen in standardisation formula. (σ= √18, 3√2, 4.2426 ≤ σ ≤ 4.243 implies correct variance) Withhold mark if variance clearly identified as standard deviation.
15. [M1] Substituting their μ and positive σ into one standardising formula (any number for 28.5 or 34.5), not σ², not √σ.
16. [M1] Using continuity corrections 27.5 or 28.5 and 34.5 or 35.5 in their 2 separate standardisation formula.
17. [M1] Appropriate area Φ, from final process. Must be a probability.
18. [A1] AWRT.