题目

Problem

In a certain country, the probability of more than $10\,\mathrm{cm}$ of rain on any particular day is 0.18, independently of the weather on any other day.

(a) Find the probability that in any randomly chosen 7-day period, more than 2 days have more than $10\,\mathrm{cm}$ of rain.

(3)

(b) For 3 randomly chosen 7-day periods, find the probability that exactly two of these periods have at least one day with more than $10\,\mathrm{cm}$ of rain.

(3)

题目中文翻译

在某个国家，任意一天的降雨量超过 $10\,\mathrm{cm}$ 的概率为 0.18，且与其他任何一天的天气相互独立。

(a) 求在任意随机选取的一个 7 天时段内，降雨量超过 $10\,\mathrm{cm}$ 的天数多于 2 的概率。

(3)

(b) 对于随机选取的三个 7 天时段，求其中恰有两个时段内至少有一天降雨量超过 $10\,\mathrm{cm}$ 的概率。

(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

(a)

解法一

思路

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一周超10cm雨日数服从二项分布，多于2取0、1、2的补集。

答题过程

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Let $R$ count days with more than $10\,\mathrm{cm}$ of rain in a seven-day period. Then $R\sim\mathrm{B}(7,0.18)$ , and

\begin{align*} \mathrm{P}(R>2)=&\,1-\bigl[(0.82)^7+7(0.18)(0.82)^6\\[4mm] &\,\hspace{2pt}+\binom72(0.18)^2(0.82)^5\bigr]\\[4mm] =&\,0.115414734\ldots\approx\boxed{0.115}. \end{align*}

(b)

解法一

思路

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每个七天时段先求至少一天超10cm的概率，再对三个时段恰两次成功作二项计算。

答题过程

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The probability that a seven-day period contains at least one such day is

p=1-(0.82)^7=0.750714529\ldots.

Using this as the success probability for each of the three periods,

\begin{align*} \mathrm{P}(\text{exactly two periods}) =&\,\binom32p^2(1-p)\\[4mm] =&\,0.421471162\ldots\approx\boxed{0.421}. \end{align*}

CIE 9709 2022 March Paper 52 Q2

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程