题目
At a cafe, customers ordering hot drinks order either tea or coffee. Of all customers ordering hot drinks, 80% order tea and 20% order coffee. Of those who order tea, 35% take sugar and of those who order coffee 60% take sugar.
(a) A random sample of 12 customers ordering hot drinks is selected.
Find the probability that fewer than 3 of these customers order coffee.
(b) (i) A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
(ii) Write down the distribution for the number of customers who take sugar from a random sample of n customers ordering hot drinks.
(c) A random sample of 10 customers ordering hot drinks is selected.
(i) Find the probability that exactly 4 of these 10 customers take sugar.
(ii) Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
(d) In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.
题目中文翻译
在一家咖啡馆，点热饮的顾客只会点茶或咖啡。 所有点热饮的顾客中，80% 点茶，20% 点咖啡。 在点茶的人中，35% 加糖；在点咖啡的人中，60% 加糖。
(a) 选取 12 位点热饮的顾客组成随机样本。
求其中少于 3 位顾客点咖啡的概率。
(b) (i) 随机选取一位点热饮的顾客。证明该顾客加糖的概率为 0.4。
(ii) 写出从随机选取的 n 位点热饮顾客中，加糖顾客人数的分布。
(c) 选取 10 位点热饮的顾客组成随机样本。
(i) 求这 10 位顾客中恰好 4 位加糖的概率。
(ii) 已知这 10 位顾客中至少 3 位加糖，求其中不超过 6 位加糖的概率。
(d) 在随机选取的 150 位点热饮顾客中，利用合适的近似方法求至少一半顾客加糖的概率。