Quartz 4

相关题目

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Question 1 — Advanced+: Supply Chain Conditional Risk

A company relies on two suppliers:

  • P(Supplier A delayed) = 0.25
  • P(Supplier B delayed) = 0.15
  • P(A ∩ B) = 0.08

If Supplier A is delayed, what is the probability Supplier B is also delayed?

📝 点击查看解析 - 使用条件概率公式：P(B|A) = P(A∩B)/P(A)。 - 在 A 延迟时，B 延迟概率上升到 32%。 - 说明供应链风险存在相关性。

Question 2 — Advanced+: Bayesian Marketing Attribution

A digital ad campaign generates clicks.

  • 70% of clicks are from genuine users, 30% are from bots.
  • Probability of purchase given genuine click = 0.2.
  • Probability of purchase given bot click = 0.01.

If a click leads to a purchase, what is probability it was from a genuine user?

📝 点击查看解析 - 贝叶斯定理应用。 - 尽管有 30% 点击来自机器人，但转化后基本可以确定是用户。 - 表明广告转化验证对评估营销效果至关重要。

Question 3 — Advanced+: Mutually Exclusive vs. Independent

In a portfolio, two projects are modeled:

  • P(A success) = 0.4
  • P(B success) = 0.5
  • P(A ∩ B) = 0.3

Are A and B independent? Are they mutually exclusive?

📝 点击查看解析 - 独立性要求 P(A∩B)=P(A)P(B)，不成立。 - 互斥性要求 P(A∩B)=0，也不成立。 - 两者既相关，又能同时发生。

Question 4 — Advanced+: Sequential Investments

A company invests in Stage 1 with success probability 0.6.
If Stage 1 succeeds, Stage 2 investment has 0.5 success probability.
If Stage 1 fails, Stage 2 is not attempted.

What is probability of overall success?

📝 点击查看解析 - Sequential model → 第二阶段依赖于第一阶段。 - 只有 60% 情况进入第二阶段。 - 最终成功概率 = 30%。

Question 5 — Advanced+: Conditional Defects

Factory has two machines:

  • Machine A produces 70% of items, defect rate = 0.05.
  • Machine B produces 30%, defect rate = 0.1.

If an item is defective, what is probability it came from Machine B?

📝 点击查看解析 - 使用贝叶斯定理。 - 尽管 B 产量少，但缺陷率高，贡献了 46% 的不良品。

Question 6 — Advanced+: Strategic Alliances

Two firms independently decide whether to form an alliance.

  • Firm A probability = 0.4
  • Firm B probability = 0.5

What is probability at least one proposes the alliance?
What is probability both agree (alliance formed)?

📝 点击查看解析 - 并集计算至少一方提出。 - 交集计算双方都提出。 - 显示合作的潜力与实际达成概率的差距。

Question 7 — Advanced+: Stock Returns Correlation

Portfolio has two stocks:

  • P(Stock A rises) = 0.6
  • P(Stock B rises) = 0.5
  • P(A ∩ B) = 0.4

Are A and B positively or negatively correlated?

📝 点击查看解析 - 对比独立情况下的联合概率与实际数据。 - 更大 → 正相关。 - 投资组合风险降低效果减弱。

Question 8 — Advanced+: Fraud Detection Conditionality

Bank monitors transactions:

  • Fraud probability = 0.02.
  • P(flag|fraud) = 0.9.
  • P(flag|not fraud) = 0.1.

If a transaction is flagged, what is probability it is fraud?

📝 点击查看解析 - 贝叶斯推断。 - 标记后真实欺诈概率 ≈ 15.5%。 - 显示“低基数问题”在欺诈检测中的影响。

Question 9 — Advanced+: Insurance Portfolio Correlation

Two risks in insurance portfolio:

  • P(Car accident) = 0.12
  • P(House fire) = 0.06
  • P(Both) = 0.02

Are these events independent?

📝 点击查看解析 - 若独立，应乘积相等。 - 实际概率更大 → 相关性正向。 - 或许因为自然灾害影响两者同时发生。

Question 10 — Advanced+: Conditional Portfolio Strategy

Investor selects stocks:

  • P(Market up) = 0.6
  • If market up, P(Stock rises) = 0.8
  • If market down, P(Stock rises) = 0.2

What is unconditional probability stock rises?

📝 点击查看解析 - 全概率公式应用。 - 股票上涨概率整体为 56%。

Question 11 — Advanced+: Marketing Channel Overlap

Survey shows:

  • 50% customers use online channel.
  • 40% use offline channel.
  • 20% use both.

What is probability a customer uses exactly one channel?

📝 点击查看解析 - 分别去掉重叠部分，再相加。 - 结果：50% 客户只使用单一渠道。

Question 12 — Advanced+: Conditional Medical Diagnosis

Disease prevalence = 0.05.
Test sensitivity = 0.95.
Test false positive rate = 0.1.

What is probability a positive test result is correct?

📝 点击查看解析 - 贝叶斯定理 → 阳性预测值 (PPV)。 - 尽管测试准确度高，但低基数使得结果只有 33.3% 有效。

Question 13 — Advanced+: Strategic Decision Tree

Firm considers expansion:

  • Success with P=0.4, payoff=+100.
  • Failure with P=0.6, payoff=−40.

What is expected value (EV)? Should expansion proceed?

📝 点击查看解析 - 决策树方法：计算期望值。 - 正收益 → 理论上应扩张。

Question 14 — Advanced+: Airline Reliability with Redundancy

An airplane has 2 independent engines.

  • Each fails with probability 0.05.
  • Plane fails if both engines fail.

What is probability plane works?

📝 点击查看解析 - 冗余系统：只要有一台发动机正常即可。 - 可靠性 99.75%。

Question 15 — Advanced+: E-commerce Conversion Funnel

Steps:

  • Visit site: 100%
  • Add to cart: 40%
  • Checkout: 60%
  • Payment success: 90%

What is overall conversion probability?

📝 点击查看解析 - 连乘法则：每一步独立条件概率。 - 转化率 = 21.6%。

Question 16 — Advanced+: Multi-Stage Hiring

Hiring pipeline:

  • Pass resume: 0.3
  • Pass interview: 0.5
  • Accept offer: 0.7

What is probability a random applicant is hired?

📝 点击查看解析 - 分阶段筛选模型。 - 最终录用率 10.5%。

Question 17 — Advanced+: Two-Market Investment

  • P(US market grows) = 0.55
  • P(China market grows) = 0.65
  • Correlation: P(both grow)=0.4

What is probability at least one grows?

📝 点击查看解析 - 使用并集公式。 - 至少一个市场增长概率 80%。

Question 18 — Advanced+: Data Center Backup

  • Primary server uptime=0.98
  • Backup server uptime=0.95
  • Independent.

What is probability system is up?

📝 点击查看解析 - 系统失效 = 主+备都挂。 - 系统可用率=99.9%。

Question 19 — Advanced+: Employee Attendance

Each employee absence probability=0.1.
What is probability at least 1 of 5 employees absent?

📝 点击查看解析 - 补集法：所有人都出勤的概率为 0.9^5。 - 至少一人缺勤 = 40.95%。

Question 20 — Advanced+: Strategic Product Decision

Firm considers two product lines:

  • Product A success=0.6, profit=50
  • Product B success=0.5, profit=70
  • P(A ∩ B)=0.3, joint profit=120

What is expected profit?

📝 点击查看解析 - 枚举所有结果：双赢、单赢、全输。 - 加权求期望。 - 最终 EV=65。

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