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I'm taking an introductory course on probability theory. We've been introduced to the function $P(E), E \subseteq S$ ($S$ being the sample space of some random process) which maps the event $E$ to its corresponding probability. Clearly, $P$ is a function from $\mathcal{P}(S)$ to $[0,1]$.

My question appears when we introduce conditional probability. By the above reasoning $P(A|B)\ A,B\subseteq S$ would imply that $A|B$ is a subset of $S$. My problem is that I cannot see how $A|B$ can be a subset of $S$. I don't even see how $A|B$ is a set, and even if it is a set I cannot visualize how it relates to $A$ and $B$. It is clearly distinct from $A\cap B$ since $P(A|B)=P(A\cap B)$ does not hold in general.

Have I misunderstood something? Is the definition of 'event' as a subset of the sample space incorrect? What is the domain of $P$?

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up vote 4 down vote accepted

I might as well post an answer. =) This is a good question and it's nice that you pay attention to the notation. First of all, there is nothing much wrong with your understanding of an event as a subset of the sample space*. The probability "function" has the set of events as its domain, and it's output is a real number in $[0,1]$.

However, when we talk of conditional expectations, we are not talking about the same function $P$, but a different (related) one. Given $B \subseteq S$ with $P(B)>0$, we define a new probability distribution, say $Q$, given by $$ Q(A) \stackrel{\text{def}}{=} \frac{P(A \cap B) }{P(B)} .$$ The domain of $Q$ is again the set of events. I will leave it as an (instructive!) exercise to show that $Q$ defines a probability distribution over $S$. We call this distribution "$P$ conditioned on $B$".

Now the point is that since the function $Q(\cdot)$ clearly depends on $P$ as well as $B$ (think of it as being indexed by $B$), we would like a notation that explicitly shows both these dependences. One choice could simply be $P_B(\cdot)$. But conventionally, this is notated as $P(⋅ \mid B)$. Remember that this notation should not be interpreted to mean that $A|B$ is a set whose probability we are calculating. Rather, we are still calculating the probability of $A$, but this time it's conditioned on $B$.

*Actually, it is wrong technically, and the "correct" definition of probability and events requires measure theory set-up. Since I presume you do not have (certainly I don't, anyway!) the background, I'll ignore this complication in this answer.

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Let's start wit this example:

The expierment is throwing a dice and observing the number on its face.

Your $S$ is the SET composed of the numbers 1,2,3,4,5,6.

Define A as the event that the number is Odd.

Hence A is the set of odd numbers: 1,3,5

Define B as the event that the number is Even.

Hence B forms a set of even numbers:2,4,6

$P(A|B)$ is the probability that you get an odd number after getting an even number. Which is a number that is a subset of A, that is it has to be one of the numbers 1,3,5.

I think the above covers your question below

how A|B is a set,

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$A\mid B$ is not a set. Nor is $A\mid B$ actually referred to in the expression $P(A\mid B)$. Rather the operation $P(\ \ \ \mid B)$, which depends on the set $B$, is applied to the set $A$.

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