题目

Problem

Kayla has a bag containing 3 red marbles, 1 blue marble and 2 green marbles. She selects one marble from the bag at random and does not replace it in the bag. She repeats this process until she obtains a green marble. The random variable $X$ is the number of marbles that she needs to select until she obtains a green marble.

(a) Draw up the probability distribution table for $X$ .

(4)

(b) Find $\operatorname{Var}(X)$ .

(3)

题目中文翻译

Kayla 有一个袋子，内有 3 颗红色弹珠、1 颗蓝色弹珠和 2 颗绿色弹珠。她从袋中随机取出一颗弹珠且不放回。她重复这个过程，直到取出一颗绿色弹珠为止。随机变量 $X$ 表示她为取出一颗绿色弹珠所需取出的弹珠数。

(a) 列出 $X$ 的概率分布表。

(4)

(b) 求 $\operatorname{Var}(X)$ 。

(3)

原卷通用作答要求

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。

解答

(a)

解法一

思路

展开

只有4颗非绿珠，所以首次绿珠至迟在第5次出现。逐次不放回抽取时，非绿珠数和总数都随抽取减少。

答题过程

展开

Combine the red and blue marbles as four non-green marbles. There are two green marbles.

\begin{align*} \mathrm{P}(X=1)=&\,\frac26=\frac5{15},\\[4mm] \mathrm{P}(X=2)=&\,\frac46\frac25=\frac4{15},\\[4mm] \mathrm{P}(X=3)=&\,\frac46\frac35\frac24=\frac3{15},\\[4mm] \mathrm{P}(X=4)=&\,\frac46\frac35\frac24\frac23=\frac2{15},\\[4mm] \mathrm{P}(X=5)=&\,\frac46\frac35\frac24\frac13\frac22=\frac1{15}. \end{align*}

Hence

$x$	$1$	$2$	$3$	$4$	$5$
$\mathrm{P}(X=x)$	$\frac5{15}$	$\frac4{15}$	$\frac3{15}$	$\frac2{15}$	$\frac1{15}$

(b)

解法一

思路

展开

使用(a)的分布求均值及二阶矩，再相减得到方差。

答题过程

展开

\begin{align*} \mathrm{E}(X)=&\,\frac{5+8+9+8+5}{15}=\frac73,\\[4mm] \mathrm{E}(X^2)=&\,\frac{5+16+27+32+25}{15}=7. \end{align*}

Therefore,

\operatorname{Var}(X)=7-\biggl(\frac73\biggr)^2 =\boxed{\frac{14}{9}}.

CIE 9709 2025 November Paper 52 Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程