题目

Problem

$30\%$ of the residents of Wimfield own an electric car. Three residents are chosen at random.

(a) Find the probability that either all three own an electric car or none of them owns an electric car.

(2)

A random sample of 125 of the residents of Wimfield is selected.

(b) Use a suitable approximation to find the probability that more than 45 of these residents own an electric car.

(5)

题目中文翻译

Wimfield 的居民中有 $30\%$ 拥有电动汽车。随机选出三名居民。

(a) 求这三人都拥有电动汽车或都不拥有电动汽车的概率。

(2)

从 Wimfield 的居民中随机抽取一个容量为 125 的样本。

(b) 使用适当的近似方法，求这些居民中有超过 45 人拥有电动汽车的概率。

(5)

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

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三人都拥有与三人都没有互斥，分别求概率后相加。

答题过程

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\mathrm{P}=(0.3)^3+(0.7)^3=0.027+0.343=\boxed{0.37}.

(b)

解法一

思路

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人数服从二项分布，用正态近似。超过45即至少46，修正边界为45.5。

答题过程

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Let $X\sim\mathrm{B}(125,0.3)$ . Since $np=37.5$ and $n(1-p)=87.5$ are both greater than 5, use $Y\sim\mathrm{N}(37.5,26.25)$ .

With a continuity correction,

\begin{align*} \mathrm{P}(X>45)=&\,\mathrm{P}(X\geq46)\\[4mm] \approx&\,\mathrm{P}(Y>45.5)\\[4mm] =&\,1-\Phi\biggl(\frac{45.5-37.5}{\sqrt{26.25}}\biggr)\\[4mm] =&\,0.0592099\ldots\\[4mm] \approx&\,\boxed{0.0592}. \end{align*}

CIE 9709 2024 November Paper 53 Q1

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程