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题干不是很明白，老师能翻译一下吗

NO.PZ2021062201000005

问题如下：

An analyst estimates that 20% of high-risk bonds will fail (go bankrupt). If she applies a bankruptcy prediction model, she finds that 70% of the bonds will receive a "good" rating, implying that they are less likely to fail. Of the bonds that failed, only 50% had a "good" rating.

Use Bayes' formula to predict the probability of failure given a "good"rating. (Hint, let P(A) be the probability of failure, P(B) be the probability of a "good" rating, P(B | A) be the likelihood of a "good" rating given failure, and P(A | B) be the likelihood of failure given a "good" rating.)

选项：

5.7%

14.3%

28.6%

解释：

B is correct. With Bayes' formula, the probability of failure given a "good"rating is：

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

where

P(A) = 0.20 = probability of failure

P(B) =0.70 = probability of a "good" rating

P(B | A) =0.50 = probability of a "good" rating given failure

With these estimates, the probability of failure given a "good" rating is：

$P(A|B) = \frac{{P(B|A)}}{{P(B)}}P(A) = \frac{{0.5}}{{0.7}} \times 0.20 = 0.143$

If the analyst uses the bankruptcy prediction model as a guide, the probability of failure declines from 20% to 14.3%.

知识点：Probability Concepts-Bayes' Formula

题目不是很明白，分不清条件了，上课的时候只对贝叶斯画图比较熟悉，单个条件下的不太熟

同学你好，

1）本题给出的是两个非条件概率，即P(fail)=20%，和P(good rating)=70%；以及一个条件概率，即P(good rating | fail)=50%。由于画图法只能解决一个非条件概率，两个条件概率的问题，这种题干设计无法通过画图解决。需要使用公式法。

2）公式法要比画图法简单的多。贝叶斯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)。

3）在实际题目中，不用去设定谁是A谁是B，题干即使给了hint也不用看，直接按照最后的问法写出来就可以。本题问的是 P(fail | good rating)，直接可以对应写出公式为：

P(fail | good rating)=P(good rating | fail) / P(good rating) ×P(fail) ，直接根据题干条件代入数字，可得 50% / 70% × 20% = 0.1429。

题干不是很明白，老师能翻译一下吗

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