To simplify the conditional probability P(S1 ∩ S2 ∩ S3 | r) when S2 and S3 are independent of r, one can utilize the property that allows conditioning on independent events. The equation P((X|Y)|Z) = P(X | (Y ∩ Z)) is suggested as a valid approach, though its acceptance may vary based on the interpretation of probability, whether through measure theory or simpler frameworks. Applying standard probability laws with the condition "| Z" can also aid in simplification, such as transforming P(A ∩ B | Z) into P((A|B)|Z) P(B | Z). The discussion encourages exploring these methods to make progress in simplifying the initial equation. Understanding the independence of events is crucial in this context.
