## Simplifying the Conditional probability

P(S1 $\cap$ S2 $\cap$ S3 | r)
How do I simplfy the above equation if S2 and S3 are independent of r ?
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 Recognitions: Science Advisor I've don't recall seeing a book that gives the probability law $P((X|Y)|Z) = P(X | (Y \cap Z) )$ but I think its true. How to argue it depends on whether you are approaching probability as measure theory or something simpler. You can also apply the usual probability laws with a condition lilke "$| Z$" tagged onto every term. For example, $P(A \cap B) = P(A|B) P(B)$ so $P(A \cap B | Z) = P( (A|B)|Z) P(B | Z)$ See if you can make progress by applying those ideas.