京都大学 情報学研究科 知能情報学専攻 2019年2月実施 専門科目 F2-2

标签：

• Kyoto-University
• Proportion-and-Base-Rate

Author

祭音Myyura (with GPT 5.5)

Description

Minister X claims that his policies for preventing motorbike accidents caused by young drivers are successful: the number of young drivers causing motorbike accidents has halved. His political opponent, Mr. Y, argues on the other hand that the proportion of motorbike accidents that are caused by young drivers has increased from 20% to 40%.

(1) Explain briefly why Minister X's and Mr. Y's assertions do not contradict with each other.

(2) $A_0$ is the initial total number of accidents (with both young and old drivers), and $A_1$ is the total number of accidents after Minister X's policies. Express $A_1$ in terms of $A_0$.

题目描述

部长 X 声称其预防年轻驾驶者摩托车事故的政策有效，因为年轻驾驶者造成的事故数减少了一半。其政治对手 Y 则称，年轻驾驶者造成的事故占全部摩托车事故的比例从 $20\%$ 上升到 $40\%$。

1. 简要说明两种说法为何并不矛盾。
2. 设政策前的总事故数（年轻与年长驾驶者合计）为 $A_0$，政策后的总事故数为 $A_1$。用 $A_0$ 表示 $A_1$。

考点

• 比例与基数效应：区分某群体事故的绝对数量与其占总量比例，并由前后比例及“减半”条件建立等式求总量变化。

Kai

Let the initial total number of accidents be $A_0$.
Since young drivers initially caused $20\%$ of all accidents, the initial number of accidents caused by young drivers was

$$0.2A_0.$$

Minister X says that this number has halved. Therefore, after the policies, the number of accidents caused by young drivers is

$$\frac{1}{2}\cdot 0.2A_0=0.1A_0.$$

On the other hand, Mr. Y says that this number is now $40\%$ of the total number of accidents. Hence,

$$0.4A_1=0.1A_0.$$

Solving this equation, we obtain

$$A_1=\frac{0.1}{0.4}A_0=\frac{1}{4}A_0.$$

Therefore,

$$\boxed{A_1=\frac{1}{4}A_0}$$

(1)

Minister X's and Mr. Y's assertions do not contradict each other because Minister X is talking about the absolute number of accidents caused by young drivers, while Mr. Y is talking about their proportion among all accidents.

The number of accidents caused by young drivers has decreased from $0.2A_0$ to $0.1A_0$, but the total number of accidents has decreased even more, from $A_0$ to $\frac{1}{4}A_0$. Thus, the proportion of accidents caused by young drivers can increase from $20\%$ to $40\%$.

(2)

$$\boxed{A_1=\frac{1}{4}A_0}$$