题目

Problem

It is known that $1\%$ of houses in a certain area have a wind turbine. A random sample of $400$ houses in this area is chosen for a survey on domestic heating. The number of houses in the sample that have a wind turbine is denoted by $X$ .

Use a suitable approximating distribution to find $\mathrm{P}(X\le 3)$ .

(3)

题目中文翻译

已知某地区有 $1\%$ 的房屋装有风力涡轮机。为调查家庭供暖情况，从该地区随机抽取 $400$ 所房屋。用 $X$ 表示样本中装有风力涡轮机的房屋数量。

用一个合适的近似分布求 $\mathrm{P}(X\le 3)$ 。

(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

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大量独立住户、很小的成功概率，可用参数 $np$ 的 Poisson 近似；至多三个包含零到三个。

答题过程

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The exact model is $X\sim\mathrm B(400,0.01)$ . Since $n=400$ is large and $p=0.01$ is small, use

X\approx\mathrm{Po}(400(0.01))=\mathrm{Po}(4).

Hence

\begin{align*} \mathrm P(X\le3) \approx&\,\mathrm e^{-4}(1+4)\\[4mm] &\,+\mathrm e^{-4}\bigg(\frac{4^2}{2!}+\frac{4^3}{3!}\bigg)\\[4mm] \approx&\,\boxed{0.433}. \end{align*}

CIE 9709 2025 June Paper 61 Q1

题目

解答

解法一

思路

答题过程