Quiz 1.5 （ 贝叶斯公式 ）

Q1 — Attendance and Exam Success（出勤与考试通过率）

Question (EN): In a business statistics course, 70% of students usually attend the review session before the midterm. Records show that 85% of those who attended passed the exam, while only 35% of those who didn’t attend passed. If a randomly chosen student passed, what is the probability that the student attended the review session?

📖 点击查看翻译

在一门商业统计课程中，70% 的学生通常会参加期中复习课。记录显示，参加复习课的学生中有 85% 通过了考试，而未参加的学生中只有 35% 通过。若随机选取一名通过考试的学生，求该学生参加复习课的概率。

📖 点击查看答案

使用贝叶斯公式： $P(\text{Attend}|\text{Pass})=\frac{P(\text{Pass}|\text{Attend})P(\text{Attend})}{P(\text{Pass})}$ 其中： $P(\text{Pass})=P(\text{Pass}|\text{Attend})P(\text{Attend})+P(\text{Pass}|\text{Not Attend})P(\text{Not Attend})$ 代入数据： $P(\text{Pass})=(0.85)(0.70)+(0.35)(0.30)=0.595+0.105=0.70$ 因此： $P(\text{Attend}|\text{Pass})=\frac{0.85(0.70)}{0.70}=0.85$

📝 点击查看解析

• 已知条件：复习课出勤率、出勤与通过率之间的条件概率。
• 求解目标：在已知“通过考试”的前提下，求“参加复习课”的后验概率。
• 思路：使用贝叶斯定理结合全概率公式计算。
• 结论：通过学生中有 85% 曾参加复习课，说明出勤对通过率影响显著。

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Q2 — Product Defect and Machine Source（产品瑕疵与机器来源）

Question (EN): A factory uses two machines, A and B, to produce parts. Machine A produces 65% of all parts, and Machine B produces 35%. The defective rate is 2% for Machine A and 6% for Machine B. If a randomly chosen part is found defective, what is the probability it was produced by Machine B?

📖 点击查看翻译

一家工厂使用两台机器 A 和 B 生产零件。A 机器生产全部零件的 65%，B 机器生产 35%。A 机器的次品率为 2%，B 机器的次品率为 6%。若随机发现一个次品，求该次品来自机器 B 的概率。

📖 点击查看答案

$P(B|\text{Defect})=\frac{P(\text{Defect}|B)P(B)}{P(\text{Defect})}$ $P(\text{Defect})=0.02(0.65)+0.06(0.35)=0.013+0.021=0.034$ $P(B|\text{Defect})=\frac{0.06(0.35)}{0.034}=0.6176$

📝 点击查看解析

• 机制：贝叶斯定理应用于源头识别问题。
• 结果解释：虽然B只生产35%的零件，但由于次品率高，约有61.8%的次品来自B机。
• 现实启示：质量控制中要结合比例与缺陷率分析源头风险。

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Q3 — Marketing Email and Purchase Conversion（营销邮件与购买转化）

Question (EN): A company found that 40% of users opened its marketing email. Among those who opened it, 30% made a purchase; among those who didn’t open, only 5% made a purchase. If a randomly selected customer made a purchase, what is the probability that they opened the email?

📖 点击查看翻译

一家公司发现有 40% 的用户会打开其营销邮件。打开邮件的用户中有 30% 进行了购买，而未打开的用户中只有 5% 进行了购买。若随机选取一名购买用户，求该用户打开邮件的概率。

📖 点击查看答案

$P(\text{Open}|\text{Buy})=\frac{0.30(0.40)}{0.30(0.40)+0.05(0.60)}=\frac{0.12}{0.12+0.03}=0.80$

📝 点击查看解析

• 这是典型的“营销转化溯源”问题。
• 尽管仅 40% 的用户打开邮件，但在购买人群中，80% 来自打开邮件者。
• 表明邮件打开率虽低，但影响购买行为显著。

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Q4 — Health Screening and Disease Detection（健康筛查与疾病检出）

Question (EN): In a population, 2% have a certain disease. The test for this disease gives a positive result in 99% of diseased individuals (true positive) and in 3% of healthy individuals (false positive). If a person’s test result is positive, what is the probability that they actually have the disease?

📖 点击查看翻译

在某人群中，2% 患有某种疾病。该疾病的检测对患病者 99% 呈阳性，对健康者 3% 误报阳性。若某人检测呈阳性，求其实际患病的概率。

📖 点击查看答案

$P(\text{Disease}|\text{Positive})=\frac{0.99(0.02)}{0.99(0.02)+0.03(0.98)}=\frac{0.0198}{0.0198+0.0294}=0.402$

📝 点击查看解析

• 虽然检测准确率高，但由于疾病基率（先验概率）低，阳性预测值仅约 40%。
• 说明即使测试很灵敏，罕见病的阳性结果仍需进一步确认。
• 体现贝叶斯思维在医学筛查中的重要性。

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Q5 — Customer Churn and Complaint Record（客户流失与投诉记录）

Question (EN): A telecom company records that 15% of customers are at risk of leaving (churning). Among those who complained, 45% eventually left, while among those who never complained, only 8% left. If a customer has left, what is the probability that they had complained before?

📖 点击查看翻译

某电信公司记录显示，15% 的客户有流失风险。曾投诉客户中 45% 最终流失，而未投诉客户中仅 8% 流失。若某客户已流失，求其曾投诉的概率。

📖 点击查看答案

$P(\text{Complain}|\text{Left})=\frac{0.45(0.15)}{0.45(0.15)+0.08(0.85)}=\frac{0.0675}{0.0675+0.068}=0.498$

📝 点击查看解析

• 约 49.8% 的流失客户曾投诉，显示投诉与流失高度相关。
• 若公司想预测潜在流失者，应重点监控投诉记录。
• 属于商业预测中“后验概率”决策模型的典型案例。

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