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：
A.5.7%
B.14.3%
C.28.6%
解释：
B is correct. With Bayes' formula, the probability of failure given a "good"rating is：
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：
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
Of the bonds that failed, only 50% had a "good" rating.这句话的意思是不是所有failed bond里面，有50%是good rating？
如果是的话，那么20% of high-risk bonds will fail (go bankrupt)，100个债券里面有20个会fail，里面50%是good，那么又是good，又fail的数量是10个。这个理解哪里有问题？
