The numbers of customers arriving at service desks A and B during a 10-minute period have the independent distributions Po(1.8) and Po(2.1) respectively.

Mark scheme

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

1. [B1] λ = 1.8 * (15/10) = 2.7
2. [M1] 1 – e⁻².⁷(1 + 2.7 + 2.7²/2!) or 1 – e⁻².⁷(1 + 2.7 +3.645) or 1- (0.06721 +0.1815 +0.2450)
3. [A1] = 0.506 (3 sf)
4. [B1] λ_A = 1.8 * (5/10) = 0.9, λ_B = 2.1 * (5/10) = 1.05. Total λ = 0.9 + 1.05 = 1.95
5. [M1] e⁻¹.⁹⁵(1 + 1.95 + 1.95²/2! + 1.95³/3!) or e⁻¹.⁹⁵(1 + 1.95 + 1.90125 +1.2358)
6. [A1] = 0.866
7. [M1] 1 – e⁻².¹x ≥ 0.90 or 1 – e⁻ᶫ ≥ 0.90
8. [M1] [e⁻².¹x < 0.1] or e⁻ᶫ < 0.1 -2.1x < ln0.1 or -λ < ln0.1 [ λ > 2.3026, 2.3026/2.1 ]
9. [*A1] 1.096 or 10.96 accept 1.097 or 10.97
10. [A1 dep] She must wait for at least 11 minutes