题目

Problem

A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4. On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day.

(a) Find the probability that it does not rain on any of the 4 days of the event.

(1)

(b) Find the probability that the first day on which it rains during the event is Tuesday.

(2)

(3)

题目中文翻译

一项体育赛事从周日开始，持续 4 天。周日下雨的概率为 0.4。此后的任意一天，若前一天下雨，则当天下雨的概率为 0.7；若前一天没有下雨，则当天下雨的概率为 0.2。

(a) 求赛事的 4 天中每天都不下雨的概率。

(1)

(b) 求赛事期间首次下雨的日子是周二的概率。

(2)

(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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周日不雨概率0.6，以后每天均在前日不雨条件下取0.8。

答题过程

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\mathrm{P}(\text{no rain on all four days})=0.6(0.8)^3=\boxed{0.3072}.

(b)

解法一

思路

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周日、周一不雨，周二雨；周三结果不受限制，因此不乘额外因子。

答题过程

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\mathrm{P}(\text{first rain on Tuesday})=0.6(0.8)(0.2)=\boxed{0.096}.

(c)

解法一

思路

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每天是否下雨并不独立，必须按唯一雨天的位置列四条不同条件概率路径。

答题过程

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Let R mean rain and D mean dry, with days ordered Sunday to Wednesday.

Pattern	Probability
RDDD	$0.4(0.3)(0.8)(0.8)=0.0768$
DRDD	$0.6(0.2)(0.3)(0.8)=0.0288$
DDRD	$0.6(0.8)(0.2)(0.3)=0.0288$
DDDR	$0.6(0.8)(0.8)(0.2)=0.0768$

Adding these disjoint cases,

\mathrm{P}=0.0768+0.0288+0.0288+0.0768=\boxed{0.2112}.

CIE 9709 2023 June 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

(c)

解法一

思路

答题过程