**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%

B.14.3%

C.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

implying that they are less likely to fail. 这句话是什么意思，干扰项吗