The question that I asked at Is the Event a Conditional Probability or an Intersection? may be similar, but I'm confused over the following question from a different textbook. I'd be shocked if two textbooks make the same mistake! It's Example 2e on P61 and 62 in A First Course in Pr, 8th ed, by Sheldon Ross.
Problem: I'm undecided as to whether to take French or chemistry. I estimate that the probability of receiving an A grade would be $\frac{1}{2}$ in French and $\frac{2}{3}$ in chemistry. If I decide based on the flip of a fair coin, what's the probability that I get an A in chemistry?
Given Solution: Let $C$ be the event that I take chemistry and $A$ denote the event that I receive an A in whatever course I take. Then the desired probability is: $Pr(A \cap C) = Pr(C)Pr(A|C) = (1/2)(2/3) $.
$ \Large 1.$ Why is the probability that I get an A in chemistry $Pr(A \cap C)$ and NOT $Pr(A|C)$? How do I decide which is right?
$ \Large 2.$ Why is $\frac{2}{3} = Pr(A|C)$ and NOT $Pr(A \cap C)$? How do I decide which is right?