# 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) $(A \cap R^C)$.

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Why must (ii) and (iii) involve $A$ and not $A^C$? Must there be a aircraft present for a false alarm? – bbnkttp Mar 25 '13 at 22:20
How is $R\mid A^c$ defined? – Stefan Hansen Mar 25 '13 at 22:21
@Sadar: Thank you. Typos fixed. Stefan Hansen: I edited my post. – Timere Mar 25 '13 at 22:25
I would have gone for (i) $R\cap A$, (ii) $R \cap A^c$, (iii) same as (ii), and (iv) $R^c \cap A$. The notation $A|B$ is new to me, and questionable at best. – copper.hat Mar 25 '13 at 22:30

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.

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Thank you for your response. Are you saying that in this example it's correct to represent the probability of (ii) by either $Pr(R|A^C)$ or $Pr(R \cap A^C)$, after which the probability of (iii) must be represented by the other notation that's left? In other words, there's no definitive choice here? – Timere Mar 25 '13 at 22:27
Basically, the language is unpleasantly vague in all four cases. – Michael Hardy Mar 25 '13 at 22:29
Thank you for your response. Since $\Pr \left(R \cap A^C \right) \neq \Pr \left(R|A^C \right)$ in general, should this example just be ignored and skipped? – Timere Mar 26 '13 at 16:00
Perhaps..... but you should make sure you know the meanings of the various notations. – Michael Hardy Mar 27 '13 at 1:09
Thank you very much. – Timere Mar 30 '13 at 12:45

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?

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