Is the Event a Conditional Probability or an Intersection?

_{My question is based on Example 1.9, p 22, *Introduction to Probability (1 Ed, 2002) by Bertsekas, Tsitsiklis.}

Define the event $A$ = {an aircraft is present} and $R$ = {the radar registers an aircraft presence}. Express the following events in terms of $A$, $R$, and/or their complements.

$\begin{align} & \text{(i) The radar correctly registers an aircraft presence}\text{.} \\ & \text{(ii) The radar falsely registers an aircraft presence}\text{.} \\ & \text{(iii) A false alarm} \\ & \text{(iv) A missed detection} \\ \end{align}$

From the definition of $A$ and $R$, I understand that both (ii) and (iii) must feature $A^C$ and $R$. However, how do I decide which is $(R | A^C )$ and which is $(R \cap A^C)$?

The textbook symbolised only (iii) and (iv):

(iii) $(R \cap A^C)$ (iv) $(A \cap R^C)$.

There's no such thing as $A\mid B$. When one writes $\Pr(A\mid B)$, one is NOT writing about the probability of something that's called $A\mid B$.

It is NOT the conditional probability of something called $A\mid B$.

Rather, it is the conditional probability given $B$, of $A$.

The distinctions that the authors are making are distinctions in their own conventions and are worth avoiding. I think that book is deficient in a number of respects. Unfortunately, books that avoid mistakes of that kind don't really call the reader's attention to things like what I wrote above, and so those mistakes persist.

A probability is of course a number. An event is not a number. An event may be an intersection of two sets. An event is never a conditional probability, since an event is not a number.

The textbook answers are wrong. They are all intersections $\bigcap$, the events are correct but this symbol $|$ is wrong. Typos like this should not make it into a published book but they often do. Care to calculate that probability?