题目

Problem

A factory makes loaves of bread in batches. One batch of loaves contains $X$ kilograms of dried yeast and $Y$ kilograms of flour, where $X$ and $Y$ have the independent distributions $\mathrm{N}(0.7,0.02^2)$ and $\mathrm{N}(100.0,3.0^2)$ respectively.

Dried yeast costs $13.50 per kilogram and flour costs $0.90 per kilogram. For making one batch of bread the total of all other costs is $55. The factory sells each batch of bread for $200.

Find the probability that the profit made on one randomly chosen batch of bread is greater than $40.

(7)

题目中文翻译

一家工厂分批制作面包。每批面包含 $X$ 千克干酵母和 $Y$ 千克面粉，其中 $X$ 和 $Y$ 相互独立，分别服从 $\mathrm{N}(0.7,0.02^2)$ 和 $\mathrm{N}(100.0,3.0^2)$ 分布。

干酵母每千克售价 $13.50$ 美元，面粉每千克售价 $0.90$ 美元。每批面包的其他制作成本合计为 $55$ 美元。工厂每批面包卖 $200$ 美元。

随机选取一批面包，求该批面包的利润超过 $40$ 美元的概率。

(7)

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

展开

先求随机原料成本的正态分布。利润大于40等价于原料成本小于105，固定收入和其他成本只影响均值，不增加方差。

答题过程

展开

Let $C=13.50X+0.90Y$ be the cost of dried yeast and flour, in dollars. By independence,

\begin{align*} \mathrm{E}(C)=&\,13.50(0.7)+0.90(100)\\[4mm] =&\,99.45, \end{align*}

and

\begin{align*} \mathrm{Var}(C)=&\,(13.50)^2(0.02)^2+(0.90)^2(3.0)^2\\[4mm] =&\,0.0729+7.29=7.3629. \end{align*}

Thus $C\sim\mathrm{N}(99.45,7.3629)$ .

The profit is $P=200-55-C=145-C$ , so

P>40\quad\Longleftrightarrow\quad C<105.

Therefore

\begin{align*} \mathrm{P}(P>40)=&\,\Phi\bigg(\frac{105-99.45}{\sqrt{7.3629}}\bigg)\\[4mm] =&\,0.979590\ldots\\[4mm] =&\,\boxed{0.980}\quad\text{(3 s.f.).} \end{align*}

CIE 9709 2023 November Paper 62 Q6

题目

解答

解法一

思路

答题过程