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Let $X,Y,Z$ be random variables jointly distributed on the same probability space then we know the conditional probability $P(X=a,Y=b\mid Z)$ is a random variable which is $Z$ measurable. Can we say

$$P(X=a,Y=b\mid Z)=P(X=a\mid Y=b,Z)P(Y=b\mid Z)?$$

Moreover is it true to say

$$\operatorname E[P(x=a,Y=b \mid Z)]=P(x=a,Y=b)?$$

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  • $\begingroup$ That last is one of the related things that are sometimes called the law of total probability. It can be stated by saying that the prior expected value of the posterior probability is equal to the prior probability. $$\operatorname E( \Pr(A\mid Z)) = \Pr(A). $$ $\endgroup$ Commented Feb 8, 2019 at 20:49
  • $\begingroup$ Thank you Michael. What can we say about the first comment? $\endgroup$
    – user131465
    Commented Feb 8, 2019 at 21:06
  • $\begingroup$ The problem with your first question is: How do you define $P(X = a ~|~ Y = b,Z)$? If you define it as $P(X = a, Y = b|Z) / P(Y = b|Z)$, then the answer is of course yes. I don't think this is what you want. Otherwise it seems more simple and clear to work with conditional expectations with respect to some sub-$\sigma$-algebras $\cal{G} \subset \cal{H}$. $\endgroup$ Commented Feb 8, 2019 at 23:29

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