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庄鑫 · 2021年12月16日

为什么没法画图讲解呢？

  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

不是说遇到贝叶斯问题就直接画图，不需要背公式吗？怎么讲师的回复说这道题没法画图了呢？

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星星_品职助教 · 2021年12月16日

同学你好，

每一个关于“没法画图”的回复里都非常清晰的解释了原因。

重述如下：

不记公式是有前提条件的。

老师讲这句话的背景是在讲画图的做法。所以准确的说法是“对于能画图的题目，不需要记公式”。能画图的题目是给出一个非条件概率画在第一支。直接画图解就可以了。

但这道题目是两个非条件概率，此时无法画图，就需要通过公式来解

naomi_jin · 2022年01月28日

老师这道题实在没看懂： 1、为什么这道题是两个非条件概率，怎么识别和判断？ 2、两个公式P（AB），P（R）在视频里没讲具体怎么对应题目中的条件 你能解释一下么？

星星_品职助教 · 2022年01月29日

@naomi_jin

同学你好，同样的问题提问一次即可。复制回复如下：

1）条件概率（如P(A|B)）和非条件概率的定义上课已经讲过。例如“ 5% of all CEOs are fired”就是无任何条件下，所有CEO被解雇的概率为5%；但“30% of all CEOs who were fired had "good" performance”就是有条件的30%，在good performance这个条件下被解雇的CEO为30%；

2）贝叶斯公式的形式和讲过的乘法法则一致；

3）题干中的Hint已经明确标识出了条件的对应。


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