题目
A sweet shop produces different coloured sweets and mixes them to sell in small, medium or large bags.
The proportion of sweets that are blue is .
Each small bag is filled with a random sample of sweets.
The mean number of blue sweets in a small bag is 3 and the variance of the number of blue sweets in a small bag is 2.55
(a) Find the value of and the value of
The proportion of sweets that are green is 0.2
Each medium bag contains a random sample of 40 sweets.
For a medium bag,
(b) find the probability that the number of green sweets exceeds the expected number of green sweets.
The proportion of sweets that are yellow is 0.1
Each large bag contains a random sample of 80 sweets.
(c) (i) Use a Poisson approximation to find the probability that a large bag of sweets contains at least 12 yellow sweets.
(ii) Justify why a Poisson approximation is suitable in this case.
(Total for Question 3 is 10 marks)
题目中文翻译
一家糖果店生产不同颜色的糖果，并将它们混合装入小、中或大袋中出售。
蓝色糖果的比例为 。
每小袋装入 颗糖果的随机样本。
一小袋中蓝色糖果的平均数为 3，方差为 2.55。
(a) 求 的值和 的值。
绿色糖果的比例为 0.2。
每中袋装入 40 颗糖果的随机样本。
对于一中袋
(b) 求绿色糖果数量超过期望数量的概率。
黄色糖果的比例为 0.1。
每大袋装入 80 颗糖果的随机样本。
(c) (i) 使用泊松近似求一大袋糖果中至少有 12 颗黄色糖果的概率。
(ii) 说明为什么泊松近似在此情况下是合适的。
（第 3 题共 10 分）