Suppose that 75% of all people with credit records improve their credit ratings within three years. Suppose that 18% of the population at large have poor credit records, and of those only 30% will improve their credit ratings within three years.
What percentage of the people who will improve their credit records within the next three years are the ones who currently have good credit ratings?
I defined $A$ as the event that a randomly selected person has a poor rating and $B$ as the event that a randomly selected person will improve their rating within three years. $$P(A) = \frac{18}{100}, P(B) = \frac{75}{100}$$ $$P(A^cB)=P(B)-P(A)=\frac{57}{100}$$ $$P(A^c|B)=\frac{P(A^cB)}{P(B)}=\frac{19}{25}$$ However, this answer is wrong. I'm not sure where the 30% statistic comes in. What am I doing wrong and what approach should I take instead?
EDIT: I was able to solve the problem with your help. Thank you! I think my reasoning for this new solution should be sound. $$P(A) = \frac{18}{100}, P(B) = \frac{75}{100}, P(B|A)=\frac{30}{100}$$ $$P(A^c|B)=1-P(A|B)=1-\frac{P(AB)}{P(B)}=1-\frac{P(A)P(B|A)}{P(B)}=\frac{116}{125}$$
Try visualizing the situation for 1.000 people.
750 people will improve their ratings.
180 people have poor ratings - meaning the remaining 820 have good ratings.
30% of the 180 people will improve - that is 54 people.
Since a total of 750 people improved that means that from the 820 people with good ratings we need 750-54=696 people to improve.
So we have 696 out of 820 people.
Try "translating" this specific (n=1000) example to probabilities.