题目

Problem

The masses, in kilograms, of small and large bags of wheat have the independent distributions $\mathrm{N}(16.0,0.4)$ and $\mathrm{N}(51.0,0.9)$ respectively.

Find the probability that the total mass of $3$ randomly chosen small bags is greater than the mass of one randomly chosen large bag.

(5)

题目中文翻译

小袋和大袋小麦的质量（以千克为单位）相互独立，分别服从 $\mathrm{N}(16.0,0.4)$ 和 $\mathrm{N}(51.0,0.9)$ 分布。

随机选取 $3$ 小袋小麦和 $1$ 大袋小麦，求 $3$ 小袋的总质量大于 $1$ 大袋质量的概率。

(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

解法一

思路

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三小袋总质量减去一大袋质量为正态变量。独立质量相加时方差相加，不能将三袋当作同一质量的三倍。

答题过程

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Let $S_1,S_2,S_3$ be the masses of the three small bags and $L$ the mass of the large bag. By independence,

D=S_1+S_2+S_3-L.

Then

\begin{align*} D\sim&\,\mathrm{N}(3(16)-51,\ 3(0.4)+0.9)\\[4mm] =&\,\mathrm{N}(-3,2.1). \end{align*}

Thus

\begin{align*} &\,\mathrm{P}(S_1+S_2+S_3>L)\\[4mm] =&\,\mathrm{P}(D>0)\\[4mm] =&\,1-\Phi\bigg(\frac{0-(-3)}{\sqrt{2.1}}\bigg)\\[4mm] =&\,0.01921697\ldots\\[4mm] =&\,\boxed{0.0192}\quad\text{(3 s.f.).} \end{align*}

CIE 9709 2024 November Paper 61 Q2

题目

解答

解法一

思路

答题过程