Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] λ = 13.5 × 20/60 = 4.5
2. [M1] 1 - e⁻⁴·⁵ (1 + 4.5 + 4.5²/2! + 4.5³/3! + 4.5⁴/4!) Allow one end error Allow any λ. Poisson expressions must be seen
3. [A1] = 0.468 (3 sf) If M0 awarded allow SC B1 for 0.468
4. [B1] λ = 13.5 × 12 = 162 (X ~ Po(162) ⇒ X ~ N(162, 162))
5. [M1] (149.5 - '162')/√'162' and (160.5 - '162')/√'162' (=-0.982 and -0.118) One of these; allow with incorrect or no continuity correction
6. [M1] Φ('0.982') - Φ('0.118') oe Area consistent with their values (both standardisations must be seen)
7. [A1] = 0.290 (3 sf) Allow 0.29
8. [M1] λ_C = 13.5 × 10/60 = 2.25, λ_T = 3.6 × 10/15 = 2.4. Total λ = 2.25 + 2.4 = 4.65. Attempt to find λ
9. [M1] e⁻⁴·⁶⁵ (4.65⁴/4! + 4.65⁵/5! + 4.65⁶/6!) Allow any λ Allow one end error Poisson terms not be seen
10. [A1] 0.494 (3 sf) If M0 allow SC B1 for 0.494