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.