Law of Total Probability in Bayes Theorem While reading along the probabilistic robotics book, I came across these set of equations which bothers me. Let's take equation 2.13 for example. 

P(Y) was expressed into a summation referred to as law of total probability. But should it be conditioned to x' only? Does not the law of total probability state? 
 P(Y) = P(Y,X) + P(Y,X')

Then why is it only conditioned to x'? 

The law of total probability basically says, if you can partition a sample space $Y$ into sets $X_1, ..., X_n$ (which can actually be countably infinite if necessary), then
$P(Y) = \sum_{n}P(Y \cap X_n) = \sum_{n}P{(Y\mid X_{n})}P(X_n)$
where the $2$nd equality holds by definition. Don't forget $P(Y|X)P(X) = P(Y \cap X)$
So,
$P(Y) = P(Y \cap X) + P(Y \cap X^c)$
is basically the law of total probability in the case where $n = 2$, since $X$ and $X^c$ form a partition of $Y$. Of course I'm assuming some things are nicely chosen (Like your $P(X) \neq 0$), but that's the gist of it.
