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Simplifying the Conditional probability

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bhathi123

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equation.jpg

Say s2, s3, s4 are independent from r.
How do I simplify the equation above?
 

Hello,
If the events s1, s2, s3, s4 are each independent from r then so is their intersection.
Therefore from the formula of independence : P(s1 and s2 and s3 and s4 | r) = P(s1|r)*P(s2|r)*P(s3|r)*P(s4|r)
Further, due to the same hypothesis the expression becomes:
P(s1 and s2 and s3 and s4 | r) = P(s1)*P(s2)*P(s3)*P(s4)

See : https://en.wikipedia.org/wiki/Conditional_independence
 
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