A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2024 · Q511 Marks

A machine puts sweets into bags at random. The numbers of lemon and orange sweets in a bag have the independent distributions Po(3.7) and Po(2.6) respectively. A bag of sweets is chosen at random.

(a)Find the probability that the number of lemon sweets in the bag is more than 2 but not more than 5.[2]
(b)Find the probability that the total number of lemon and orange sweets in the bag is less than 4.[3]
(c)10 bags of sweets are chosen at random. Use approximating distributions to find the probability that the total number of lemon sweets in the 10 bags is less than the total number of orange sweets in the 10 bags.[6]

Mark scheme

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

e^-3.7(3.7³/3! + 3.7⁴/4! + 3.7⁵/5!) = e^-3.7(8.44217 + 7.80900 + 5.77866) = 0.20872 + 0.19307 + 0.14287
= 0.545 (3 sf)
[λ] = 6.3
e^-6.3(1 + 6.3 + 6.3²/2! + 6.3³/3!) = e^-6.3(1 + 6.3 + 19.845 + 41.6745) = 0.0018363+0.011569 +0.0364415 +0.076527
= 0.126 (3 sf)
L~N(37, 37), O~N(26, 26)
(O-L)~N(-11, 63)
For var = 37+26 SOI.
(0-'-11') / √'63' [=1.386] or (0+0.5-'-11') / √'63' [=1.449]
1 – Ф('1.386') or 1 – Ф('1.449')
= 0.0828 or 0.0829 (3 sf) or = 0.0737 or 0.0736 (3 sf)

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.