The masses, in kilograms, of large and small sacks of flour have the distributions N(55, 3²) and N(27, 2.5²) respectively.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [B1] For expected total mass `E(T) = 3×55 + 6×27 = 327` (OE, accept unsimplified).
2. [B1] For variance of total mass `Var(T) = 3×3² + 6×2.5² = 64.5` (accept unsimplified).
3. [M1] For standardizing the value: `(340 - 327) / √64.5`. Must have √.
4. [M1] For finding the correct probability area: `P(Z < 1.619) = Φ(1.619)`. Correct probability area consistent with their working.
5. [A1] Final probability `0.947 (3 sf)`.
6. [B1] For expected difference `E(L - S1 - S2) = 55 – 2 × 27 = 1` (OE e.g. `E(S1+S2 – L) = −1`. Accept unsimplified).
7. [B1] For variance of difference `Var(L - S1 - S2) = 3² + 2 × 2.5² = 21.5` (accept unsimplified).
8. [M1] For standardizing: `(0 - 1) / √21.5`. Standardising with their values. Must come from a combination attempt.
9. [M1] For finding correct probability area: `P(L - S1 - S2 > 0) = Φ('0.216')`. Correct probability area consistent with their working.
10. [A1] Final probability `0.586` or `0.585 (3 sf)`.