Hanna buys 12 hollow chocolate eggs that each contain a sweet. The eggs look identical but Hanna knows that 3 contain a red sweet, 4 contain an orange sweet and 5 contain a yellow sweet. Each of Hanna's three children in turn randomly chooses and eats one of the eggs, keeping the sweet it contained.

Mark scheme

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

1. M1 Either 12 × 11 × 10 in denominator or a × (a-1) × ( a − 2), a = 5, 4, 3 in numerator seen in at least one expression.
2. A1 One expression ax(a-1)x(a-2)/(12x11x10) , a = 5, 4, 3 (consistent in expression). Correct order of values in the numerator is essential.
3. M1 (a/d x (a-1)/e x (a-2)/f) + (b/d x (b-1)/e x (b-2)/f) + (c/d x (c-1)/e x (c-2)/f) , either d = 11, e = 10 or d = 12, e = 12. [5/12 * 4/11 * 3/10 + 4/12 * 3/11 * 2/10 + 3/12 * 2/11 * 1/10]
4. A1 0.06818. Dependent only upon the second M mark. [90/1320 or 3/44 or 0.0682]
5. M1 Either ¹²C₃ in denominator or ᵃC₃ in numerator seen in at least one expression.
6. A1 One expression ᵃC₃ / ¹²C₃ a = 5, 4, 3
7. M1 (⁵C₃ + ⁴C₃ + ³C₃) / ¹²C₃
8. A1 0.06818. Dependent only upon the second M mark.
9. M1 their P(YYY) or 60/1320 or 1/22 / their7(a) or 90/1320 or 3/44
10. A1 OE [2/3 or 0.667]
11. B1 P(^O) = 8/33 or P(^^O) = 28/165. Accept unsimplified.
12. M1 3 correct scenarios added, with at least one 3-term product of f/g x h/d x e seen, either d = 11, e = 10 or d = 12, e = 12 with correct numerator.
13. A1 0.745 < p < 0.74545 If M0 scored SC B1 0.745 < p < 0.74545. [984/1320 or 41/55 or 0.745]