题目

Problem

Students applying to Drydale College take an entrance test. A student is either accepted or rejected or required to take another test with probabilities $0.3$ , $0.2$ and $0.5$ respectively. When a student takes a second test the outcomes and probabilities are exactly the same as for the first test. A student who has to take a third test is accepted with probability $0.25$ and rejected with probability $0.75$ .

(a) Draw a tree diagram to illustrate this information, showing all the probabilities.

(2)

(b) Find the probability that a randomly chosen student who applies to Drydale College is accepted.

(2)

(c) Find the probability that a randomly chosen student who applies to Drydale College takes at least two tests given that the student is accepted.

(3)

Three friends apply to Drydale College.

(d) Find the probability that all three are rejected.

(2)

题目中文翻译

申请 Drydale College 的学生要参加入学测试。一名学生可能被录取、被拒绝或被要求再参加一次测试，概率分别为 $0.3$ 、 $0.2$ 和 $0.5$ 。当学生参加第二次测试时，可能的结果及其概率与第一次测试完全相同。必须参加第三次测试的学生被录取的概率为 $0.25$ ，被拒绝的概率为 $0.75$ 。

(a) 画出树状图来表示这些信息，并标出所有概率。

(2)

(b) 随机选取一名申请 Drydale College 的学生，求该学生被录取的概率。

(2)

(3)

三位朋友申请 Drydale College。

(d) 求三人全部被拒绝的概率。

(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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第一次和第二次均有录取、拒绝、重考三分支；只有重考才继续到下一层，第三次则只剩录取或拒绝。

答题过程

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Write A for accepted, R for rejected and T for test again.

At the first test, the probabilities are A 0.3, R 0.2 and T 0.5. After T, the second test has the same three probabilities. After a second T, the third test has A 0.25 and R 0.75.

Stop each branch as soon as the student is accepted or rejected.

(b)

解法一

思路

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录取路径只有首次录取、重考一次后录取、重考两次后录取，三条互斥路径相加。

答题过程

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\begin{align*} \mathrm{P}(\text{accepted}) =&\,0.3+0.5(0.3)+(0.5)^2(0.25)\\[4mm] =&\,0.3+0.15+0.0625\\[4mm] =&\,\boxed{0.5125}. \end{align*}

(c)

解法一

思路

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在已录取的条件下，至少两次考试表示排除首次就录取。分子保留第二或第三次录取的两条路径。

答题过程

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The probability of taking at least two tests and being accepted is

0.5(0.3)+(0.5)^2(0.25)=0.2125.

Hence

\begin{align*} &\,\mathrm{P}(\text{at least two tests}\mid\text{accepted})\\[4mm] =&\,\frac{0.2125}{0.5125}\\[4mm] =&\,\boxed{\frac{17}{41}}. \end{align*}

(d)

解法一

思路

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每人最终拒绝的概率是录取概率的补集，三位朋友的结果按独立申请连乘。

答题过程

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Each friend’s probability of rejection is $1-0.5125=0.4875$ . Therefore,

\begin{align*} \mathrm{P}(\text{all three rejected}) =&\,(0.4875)^3\\[4mm] =&\,0.115857421875\\[4mm] \approx&\,\boxed{0.116}\quad\text{(3 s.f.)}. \end{align*}

CIE 9709 2025 June Paper 55 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

(c)

解法一

思路

答题过程

(d)

解法一

思路

答题过程