The probability that a driver passes an advanced driving test is 0.3 on any given attempt. Dipak keeps taking the test until he passes. The random variable X denotes the number of attempts required for Dipak to pass the test.

Mark scheme

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

1. [M1] 1 - 0.7ⁿ seen, n = 5, 6.
2. [A1] www 0.582351 to at least 3SF.
3. [M1] Sum of first 4 or 5 correct terms no incorrect terms.
4. [A1] www 0.582351 to at least 3SF.
5. [B1] Condone 3.33, 3.3 or 10/3 – NOT 1/0.3.
6. [M1] One term seen ⁵Cₓ (p)ˣ (1-p)⁵⁻ˣ, 0 < p < 1, x ≠ 0, 5.
7. [A1] Correct expression, accept unsimplified, no terms omitted leading to final answer.
8. [B1] 0.16308 to at least 3SF.
9. [M1] One term seen ⁵Cₓ (p)ˣ (1-p)⁵⁻ˣ, 0 < p < 1, x ≠ 0, 5.
10. [A1] Correct expression, accept unsimplified, no terms omitted leading to final answer.
11. [B1] 0.16308 to at least 3SF.
12. [B1] 22.5, 22 1/2 and 15.75, 15 3/4 seen, allow unsimplified. (σ = 3√7/2 or 3.9686269... to at least 3SF implies correct variance)
13. [M1] Substituting their μ and σ into ±standardisation formula (any number for 20.5), not σ² not √o.
14. [M1] Using continuity correction 19.5 or 20.5 in their standardisation formula.
15. [M1] Appropriate area Φ, from final process, must be a probability. Expect final answer > 0.5. Note: correct final answer implies this M1.
16. [A1] 0.6925 < p ≤ 0.693