A factory makes loaves of bread in batches. One batch of loaves contains X kilograms of dried yeast and Y kilograms of flour, where X and Y have the independent distributions N(0.7, 0.02²) and N(100.0, 3.0²) respectively. Dried yeast costs $13.50 per kilogram and flour costs $0.90 per kilogram. For making one batch of bread the total of all other costs is $55. The factory sells each batch of bread for $200. Find the probability that the profit made on one randomly chosen batch of bread is greater than $40.

Mark scheme

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

1. [B1] One of these soi – can be given at early stage. Cost of dried yeast and flour: $D and $F, E(D) = 13.5 × 0.7 = 9.45, E(F) = 0.9 × 100 = 90
2. [B1] One of these soi – can be given at early stage. Var(D) = 0.02² × 13.50² = 0.0729, Var(F) = 3.0² × 0.90² = 7.29
3. [M1] Attempt to combine their D and their F with or without 55 and 200 (but variance must not include 55 or 200). Or N(99.45 +55, (0.02² × 13.50² + 3.0² × 0.90²)). Or N(200-55 -99.45, (0.02² × 13.50² + 3.0² × 0.90²)). Or N(99.45 +c, (0.02² × 13.50² + 3.0² × 0.90²)). Total cost: T~N(99.45, (0.02² × 13.50² + 3.0² × 0.90²))
4. [A1] Or N(154.45,7.3629) or N(45.55, 7.3629). Accept 3sf (accept 45.5 or 45.6). N(99.45, 7.3629) accept 99.4 or 99.5
5. [M1] 160–154.45/√7.3629 or 40–45.55/√7.3629. No mixed methods. [P(profit > $40) = P(T < 105)] (105-'99.45')/√'7.3629' [= 2.045]
6. [M1] For area consistent with their working. [P(T < 105) = P(z < 2.045) =] Φ(2.045)
7. [A1] = 0.9795 or 0.9796 or 0.98(0) or 0.979 (3 sf)