题目

Problem

Emma needs to choose one person at random from three people, $P$ , $Q$ and $R$ . She plans to throw two fair coins and note the number, $n$ , of heads. If $n$ is $0$ , she will choose $P$ . If $n$ is $1$ , she will choose $Q$ . If $n$ is $2$ , she will choose $R$ .

(a) By considering probabilities, show that the choice made by this method is not random.

(2)

Later, Emma has to choose two people at random from three people.

(b) Describe how Emma could use a single throw of a fair six-sided dice to make this random choice.

(2)

题目中文翻译

Emma 需要从 $P$ 、 $Q$ 、 $R$ 三人中随机选出一人。她打算抛掷两枚均匀的硬币，并记录正面朝上的枚数 $n$ 。如果 $n$ 为 $0$ ，她就选 $P$ 。如果 $n$ 为 $1$ ，她就选 $Q$ 。如果 $n$ 为 $2$ ，她就选 $R$ 。

(a) 通过考察概率，证明用这种方法作出的选择不是随机抽选。

(2)

后来，Emma 需要从三人中随机选出两人。

(b) 描述 Emma 如何只掷一次均匀的六面骰子来完成这次随机选择。

(2)

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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两枚硬币的四种有序结果等可能，但正面数为一有两种结果，不能把三个正面数当作等可能。

答题过程

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The equally likely outcomes are HH, HT, TH and TT. Therefore

\begin{align*} \mathrm P(n=0)=&\,\frac14,\\[4mm] \mathrm P(n=1)=&\,\frac12,\\[4mm] \mathrm P(n=2)=&\,\frac14. \end{align*}

Thus $P,Q,R$ are selected with probabilities $1/4,1/2,1/4$ respectively. They do not have equal chances, so this method does not give the required random choice among the three people.

(b)

解法一

思路

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要选两人，就把三个无序人数组合作为三个等可能结果，每组对应骰子的两个点数。

答题过程

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Throw the fair die once. Choose the pair $PQ$ for a score of $1$ or $2$ , the pair $QR$ for $3$ or $4$ , and the pair $RP$ for $5$ or $6$ .

Each possible pair is selected with probability $2/6=1/3$ , so the choice is random.

CIE 9709 2025 June Paper 63 Q4

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程