勇往直前 · 2022年04月02日

完全看不懂怎么算的，又不能画图，转晕了，请老师帮忙，否则得放弃这块了。。。

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：

1.5%

2.5%

3.0%

解释：

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

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

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：

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

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

完全看不懂怎么算的，又不能画图，转晕了，请老师帮忙，否则得放弃这块了。。。

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星星_品职助教 · 2022年04月02日

同学你好，

公式法是很简单的，有固定的流程。

①首先记忆公式，贝叶斯题目的公式和乘法法则的形式是一样的。乘法法则公式为P(AB)=P(A|B)×P(B)=P(B|A)×P(A)。把上述连等式的后两项重新组合一下，就可以得到P(A|B)=P(B|A) / P(B) × P(A)。

②在实际题目中，都不用去特意设定谁是A谁是B，直接按照最后的问法写出来就可以。本题问的是 the probability that a CEO will be fired given "good" performance，即P(CEO is fired | good rating)，据此就直接可以写出公式为：

P(CEO is fired | good rating)=P(good rating | CEO is fired) / P(good rating) × P(CEO is fired)

③此后就从题干中找条件代数就行，公式右边的三项都直接给出了。

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在本题中，找到：

1）5% of all CEOs are fired，这就是P(CEO is fired)=5%；

2）50% of stocks have above-average returns or "good" performance，这就是P(good rating)=50%；

3）30% of all CEOs who were fired had "good" performance，这就是P(good rating | CEO is fired) =30%；

代入公式，直接得到 30% / 50% × 5%=3%

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只要掌握了上述思维，贝叶斯的公式法就是纯代数字的题目。

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Avalon · 2022年05月05日

谢谢老师，讲得非常好！

星星_品职助教 · 2022年05月06日

@鲤哩哩哩哩哩

学习加油~

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完全看不懂怎么算的，又不能画图，转晕了，请老师帮忙，否则得放弃这块了。。。

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