Simplifying the Conditional probability

[bhathi123]

P(S₁ $\cap$ S₂ $\cap$ S₃ | r)
How do I simplfy the above equation if S₂ and S₃ are independent of r ?

 

[Stephen Tashi]

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.