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Clarification needed!

let A and B be two events such that P(B|A)=P(B|A^c), A^c is A complement. Are A and B independent? Please Give reasons!

Note by Sourav Agarwal
2 years, 6 months ago

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For $$P(A) \in (0,1)$$, we can write $$\displaystyle P(B|A) = \frac{P(B \cap A)}{P(A)}$$ and $$\displaystyle P(B|A^c) = \frac{P(B \cap A^c)}{P(A^c)}$$ (Using Bayes' Theorem ).

Since, $$\displaystyle P(B|A) = P(B|A^c)$$, therefore, $$\displaystyle \frac{P(B \cap A)}{P(A)} = \frac{P(B \cap A^c)}{P(A^c)}$$ $$\displaystyle \Rightarrow P(B \cap A)P(A^c) = P(A)P(B \cap A^c) \Rightarrow P(B \cap A)( 1- P(A)) = P(A) (P(B) - P(B \cap A) )$$ Simplifying, we obtain, $$\displaystyle P(A \cap B) = P(A)P(B)$$ which is sufficient to prove that $$A$$ and $$B$$ are independent. · 2 years, 6 months ago