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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?

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I read

I estimate that the probability of receiving an A grade would be $\tfrac12$ in French and $\tfrac23$ in chemistry

as if you were to choose French or chemistry, what would then be the probability of getting an A grade. Let me try to explain it a bit better. I imagine the following setup $$ \text{Coin-flip}\;\rightarrow\; \text{Picking course based on coin-flip}\;\rightarrow\;\text{Getting grade} $$ where you at first toss a coin, and based on that coin you choose between French ($\text{coin}=\text{heads}$) and chemistry ($\text{coin}=\text{tails}$) and at the end you receive your grade.

Let us suppose we haven't done anything yet (which is fair to assume, since the outcome of the coin-toss hasn't been mentioned). Then we might ask what is the probability of getting an A grade in the chemistry course. For this to happen you would have to see a coin-toss coming up tails, which is the event $C$, and you would have to receive an A grade after finishing the course.

So $P(A\cap C)$ is the probability of you getting an A in chemistry if you decide to choose between chemistry and French according to a coin-flip. On the other hand, $P(A\mid C)$ is the probability of you getting an A given that you know the coin-toss came up tails (which of course you don't know - you're at the start of the diagram above!)

This should explain why $P(A\cap C)$ is the probability you're interested in and also why $P(A\mid C)=\tfrac23$. If you wanted $P(A\cap C)=\tfrac23$ then it should have had a different wording. Maybe something along the lines of:

I estimate the probability of choosing the chemistry course and at the same time receiving an A grade would be $\tfrac23$.

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