题目

Problem

A game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn. A player who obtains a 4 in their turn scores 1 point.

(a) Show that the probability that a player obtains a 4 in one turn is $\frac{37}{64}$ .

(2)

Xeno and Yao play this game.

(b) Find the probability that neither Xeno nor Yao score any points in their first two turns.

(1)

Find the probability that Xeno scores 2 more points than Yao.

(3)

题目中文翻译

一个双人游戏使用一颗各面编号为 1、2、3、4 的均匀四面骰子。每回合重复掷骰，最多掷三次。一旦掷出 4，该回合便不再掷骰。玩家在自己的回合中掷出 4，就得 1 分。

(a) 证明玩家在一个回合中掷出 4 的概率为 $\frac{37}{64}$ 。

(2)

Xeno 和 Yao 玩这个游戏。

(b) 求 Xeno 和 Yao 在各自的前两个回合中均未得分的概率。

(1)

求 Xeno 比 Yao 多得 2 分的概率。

(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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每回合失败必须三次均非4；成功概率为其补集。

答题过程

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The probability of no 4 in all three permitted throws is $\bigl(\frac34\bigr)^3$ . Therefore

\mathrm{P}(\text{score 1 point})=1-\biggl(\frac34\biggr)^3 =1-\frac{27}{64}=\boxed{\frac{37}{64}}.

解法二

思路

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也可按首次4在第1、2、3次分类。各路径都遵守掷出4即停止。

答题过程

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A player scores if the first 4 occurs on throw 1, 2 or 3. Hence

\mathrm{P}=\frac14+\frac34\frac14+\biggl(\frac34\biggr)^2\frac14 =\frac{16+12+9}{64}=\boxed{\frac{37}{64}}.

(b)

解法一

思路

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两人各两回合，共四个独立失败回合，每回合失败概率27/64。

答题过程

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For neither player to score in either of their first two turns, all four turns must fail. Thus

\mathrm{P}=\biggl(\frac{27}{64}\biggr)^4 =0.0316763\ldots\approx\boxed{0.0317}.

(c)

解法一

思路

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每人最多3分，Xeno多2分只有(2,0)或(3,1)两种。两人的回合得分分别服从二项分布。

答题过程

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Write $p=\frac{37}{64}$ and $q=\frac{27}{64}$ . The scores $X$ and $Y$ are independent, each with distribution $\mathrm{B}(3,p)$ .

The required score pairs are $(X,Y)=(2,0)$ and $(3,1)$ . Therefore

\begin{align*} \mathrm{P}(X-Y=2) =&\,\bigl(3p^2q\bigr)q^3+p^3\bigl(3pq^2\bigr)\\[4mm] =&\,3p^2q^4+3p^4q^2\\[4mm] =&\,0.0914066\ldots\\[4mm] \approx&\,\boxed{0.0914}. \end{align*}

CIE 9709 2024 June Paper 51 Q4

题目

解答

(a)

解法一

思路

答题过程

解法二

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