题目

Problem

Marco has four boxes labelled $K$ , $L$ , $M$ and $N$ . He places them in a straight line in the order $K$ , $L$ , $M$ , $N$ with $K$ on the left. Marco also has four coloured marbles: one is red, one is green, one is white and one is yellow. He places a single marble in each box, at random. Events $A$ and $B$ are defined as follows.

$A$ : The white marble is in either box $L$ or box $M$ .

$B$ : The red marble is to the left of both the green marble and the yellow marble.

Determine whether or not events $A$ and $B$ are independent.

(3)

题目中文翻译

Marco 有四个盒子，分别标为 $K$ 、 $L$ 、 $M$ 和 $N$ 。他将盒子按 $K$ 、 $L$ 、 $M$ 、 $N$ 的顺序排成一条直线， $K$ 位于最左侧。Marco 还有四颗彩色弹珠，红色、绿色、白色和黄色各一颗。他随机地在每个盒子中放入一颗弹珠。事件 $A$ 和 $B$ 定义如下。

$A$ ：白色弹珠在盒子 $L$ 或盒子 $M$ 中。

$B$ ：红色弹珠位于绿色弹珠和黄色弹珠两者的左侧。

判断事件 $A$ 和 $B$ 是否相互独立。

(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

解法一

思路

展开

四颗不同弹珠共有24种等概率排列。先单独数A、B，再列交集的四种排列，不能先假设独立。

答题过程

展开

The white marble is equally likely to occupy any box, so

\mathrm{P}(A)=\frac24=\frac12.

Among red, green and yellow, each is equally likely to be leftmost. Hence

\mathrm{P}(B)=\frac13.

There are $4!=24$ equally likely colour orders in boxes K, L, M, N. The orders satisfying both events are

\mathrm{RWGY},\quad\mathrm{RWYG},\quad\mathrm{RGWY},\quad\mathrm{RYWG}.

Thus

\mathrm{P}(A\cap B)=\frac4{24}=\frac16 =\frac12\frac13=\mathrm{P}(A)\mathrm{P}(B).

Therefore A and B are independent.

CIE 9709 2023 March Paper 52 Q5

题目

解答

解法一

思路

答题过程