题目

Problem

A point is to be randomly plotted on the x-axis, where the units are measured in cm. The random variable $R$ represents the x coordinate of the point on the x-axis and $R$ is uniformly distributed over the interval $[-5, 19]$ A negative value indicates that the point is to the left of the origin and a positive value indicates that the point is to the right of the origin.

(a) Find the exact probability that the point is plotted to the right of the origin.

(b) Find the exact probability that the point is plotted more than 3.5 cm away from the origin.

(1)

(2)

Three independent points with x coordinates $R_1$ , $R_2$ and $R_3$ are plotted on the x-axis.

(d) Find the exact probability that

(i) all three points are more than 10 cm from the origin

(ii) the point furthest from the origin is more than 10 cm from the origin.

(3)

(2)

(Total for Question 3 is 10 marks)

题目中文翻译

一个点要随机绘制在 $x$ 轴上，单位为厘米。随机变量 $R$ 表示该点的 $x$ 坐标，且 $R$ 在区间 $[-5, 19]$ 上均匀分布。负值表示点在原点左侧，正值表示点在原点右侧。

(a) 求该点位于原点右侧的精确概率。

(b) 求该点离原点超过 3.5 cm 的精确概率。

三个独立点的 $x$ 坐标分别为 $R_1$ 、 $R_2$ 和 $R_3$ ，都被绘制在 $x$ 轴上。

(d) 求以下事件的精确概率：

(i) 三个点都离原点超过 10 cm，

(ii) 离原点最远的点超过 10 cm。

（第 3 题共 10 分）

IAL 2022 June S2 Q3

题目

解答