Simplifying the Conditional probability

bhathi123

P(S₁ [itex]\cap[/itex] S₂ [itex]\cap[/itex] 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
[itex] P((X|Y)|Z) = P(X | (Y \cap Z) ) [/itex]
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 "[itex]| Z[/itex]" tagged onto every term. For example,
[itex] P(A \cap B) = P(A|B) P(B) [/itex]
so
[itex] P(A \cap B | Z) = P( (A|B)|Z) P(B | Z) [/itex]

See if you can make progress by applying those ideas.