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苏·Xu · 2022年06月23日

问

  1. CFA I
  2. Quantitative

NO.PZ2021062201000006

问题如下：

In a typical year, 5% of all CEOs are fired for "performance" reasons. Assume that CEO performance is judged according to stock performance and that 50% of stocks have above-average returns or "good" performance. Empirically, 30% of all CEOs who were fired had "good" performance.

Using Bayes' formula, what is the probability that a CEO will be fired given "good" performance? (Hint, let P(A) be the probability of a CEO being fired, P(B) be the probability of a "good" performance rating, P(B I A) be the likelihood of a "good" performance rating given that the CEO was fired, and P(A I B) be the likelihood of the CEO being fired given a "good" performance rating.)

选项：

A.

1.5%

B.

2.5%

C.

3.0%

解释：

C is correct. With Bayes' formula, the probability of the CEO being fired given a "good" rating is：

P(A∣B)=P(B∣A)P(B)P(A)P(A|B) = \frac{{P(B|A)}}{{P(B)}}P(A)

where

P(A) = 0.05 = probability of the CEO being fired

P(B) = 0.50 = probability of a "good" rating

P(B | A) = 0.30 = probability of a "good" rating given that the CEO is fired

With these estimates, the probability of the CEO being fired given a "good" rating is：

P(A∣B)=P(B∣A)P(B)P(A)=0.30.5×0.05=0.03P(A|B) = \frac{{P(B|A)}}{{P(B)}}P(A) = \frac{{0.3}}{{0.5}} \times 0.05 = 0.03

Although 5% of all CEOs are fired, the probability of being fired given a "good" performance rating is 3%.

知识点：Probability Concepts-Bayes' Formula

为什么是2个非条件概率呢？

  1. 表现好与不好是非条件概率，非条件概率应该放在第一支；
  2. 在股票表现好的情况下被解雇的概率是30%，这个不就是条件概率嘛？应该放在第二支。


请问以上有什么问题呢？麻烦详解一下，谢谢。

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1 个答案

星星_品职助教 · 2022年06月23日

同学你好，

按照上述思路是无法画出图像的，在画图的第二步会缺少“股票表现不好的条件下，被解雇的概率”这一枝。也就无法得出答案。

可以对比本题和课上讲的例题的区别。例题是给出一个非条件概率（第一支）与两个条件概率（第二支），这种情况下才可以画图。

本题是两个非条件概率（另一个非条件概率为“ 50% of stocks have ..... "good" performance”）和一个条件概率，所以无法画图，需要通过贝叶斯的公式来解题。


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