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wyrw · 2024年08月05日

条件概率和非条件概率

NO.PZ2022123001000059

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:

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%.

为何20%是非条件概率P(A) = 0.20 = probability of failure,而不是条件概率(在high-risk Bond中20%会fail)

3 个答案

品职助教_七七 · 2024年08月09日

嗨,从没放弃的小努力你好:


已经回复了两次,“这道题里没有low-risk bond或其他的类型,所有的bond都是high risk的”。再重述一遍,这道题里的high risk bond就是全部bond,没有“high-risk的基础上”这种说法。也没有“20%是有一个条件在前”这种情况。high-risk这个描述在这道题里没有任何作用。所谓“针对全部债券”就是针对全部high risk债券。

也就是全部bond中有20%会fail;全部bond里有70%是good rating。这是两个非条件概率。high-risk不是前置条件。

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品职助教_七七 · 2024年08月08日

嗨,爱思考的PZer你好:


1)题目没有给出high risk=“un-good rating”;

2)已经回复了本题所有的bond都是high risk bond。从下文可以看出,good rating 和“un-good rating”都是在(high risk )bond中再次划分的。high risk和“un-good rating”明显不同。

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品职助教_七七 · 2024年08月06日

嗨,从没放弃的小努力你好:


这道题里没有low-risk bond或其他的类型,所有的bond都是high risk的。也就是说这并不是一个可以用来做区分的条件,有没有这个描述都不影响解题。

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加油吧,让我们一起遇见更好的自己!

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