一个袋子里有 10 颗弹珠，其中 4 颗是红色的，6 颗是蓝色的。从袋中随机不放回地抽取 4 颗弹珠。随机变量 $X$ 表示抽到的蓝色弹珠的颗数。

(a) 证明 $\mathrm{P}(X = 2) = \frac{3}{7}$。

(b) 列出 $X$ 的概率分布表。

(c) 已知抽到至少 1 颗红色弹珠和至少 1 颗蓝色弹珠，求抽到至少 2 颗蓝色弹珠的概率。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

四颗中恰有两蓝，意味着从6蓝中选2、从4红中选2，再除以全部四颗选法。将给定结果自然化简出来。

There are $\binom{10}{4}=210$ equally likely selections. To obtain exactly two blue marbles, choose two of the six blues and two of the four reds. Thus

也可以按不放回的次序计算两蓝两红。先指定蓝蓝红红的次序，再乘两蓝在四次中可占的位置数。

For the ordered colour sequence BBRR, the probability is

There are $\binom42=6$ possible positions for the two blues, each with the same probability. Hence

X可以是0至4；对每个值，从蓝珠中选X颗、红珠中选四减X颗。

For $x=0,1,2,3,4$,

The numerators are respectively $1,24,90,80,15$, giving

The numerators sum to $210$, so the probabilities sum to 1.

已知至少一红一蓝，排除全红与全蓝，所以条件范围只有X为1、2、3。其中至少两蓝的情况只有2、3。

Let $E$ be the event of at least one marble of each colour. Then $E$ is $1\leq X\leq3$.