What is the conditional probability of a product of probabilities? More precisely, suppose that $P(C) = P(A)P(B)$, where $A$ and $B$ are any two events which are not necessarily mutually independent. Consider another event $X$ (not necessarily independent of $A$ and/or $B$).
What is $P(C|X)$?
Would it be incorrect to say that it is simply: $P(C|X)=P(A|X)P(B|X)$?
Any help would be appreciated. Thank you!
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To bring some meaning to the calculations, let's roll a (fair, six-sided) die and say:
Then $P(A)=\frac{1}{6}=\frac{1}{2}\times\frac{1}{3}=P(B)P(C)$.
If $X=A$ (i.e. we roll the die and it's a $5$) then $P(A|X)=1\neq 0\times 0=P(B|X)P(C|X)$.
When we put it this way, it seems rather odd that we would expect the statement to be true: that two probabilities multiply together to give a different probability says nothing about the relation between the events.
However, when looking just at the formulae, we are used to seeing $P(A)=P(B)P(C)$ in the context that $A$ is the combination of two independent events $B$ and $C$ both happening, when the formula $P(A|X)=P(B|X)P(C|X)$ does hold). If we changed $A$ to be the event that the die lands on $6$ then we can see that this would work for our example above.
I suppose $A,B,C,X$ are all events and not random variables. Consider $(0,1)$ with Lebesgue measure and take $A=(0,1), B=(0,\frac 1 2), C=(\frac 1 2,1)$ and $X=C$ to get a counter-example.