题目
A bag contains 50 counters, each with one of the numbers 4, 7 or 10 written on it in the ratio 2 : 3 : 5 respectively.
A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as D₁ and D₂
The random variable M represents the mean of D₁ and D₂
(a) Show that P(M = 4) = 9/245
(b) Find the sampling distribution of M
A random sample of n sets of 2 counters is taken. The random variable T represents the number of these n sets of 2 counters that have a mean of 4
Given that each set of 2 counters is replaced after it is drawn,
(c) calculate the minimum value of n such that P(T = 0) < 0.15
(Total for Question 4 is 10 marks)
题目中文翻译
一个袋子里有 50 个筹码，每个筹码上写有数字 4、7 或 10，比例分别为 2 : 3 : 5。
从袋中随机抽取 2 个筹码。2 个筹码上的数字分别记录为 D₁ 和 D₂。
随机变量 M 表示 D₁ 和 D₂ 的均值。
(a) 证明 P(M = 4) = 9/245
(b) 求 M 的抽样分布。
随机抽取 n 组 2 个筹码。随机变量 T 表示这 n 组中均值为 4 的组数。
已知每组 2 个筹码在抽取后被放回，
(c) 计算最小的 n 值，使得 P(T = 0) < 0.15
（第 4 题共 10 分）