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!

#Probability

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`2^{34}`	$2^{34}$
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Comments

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.

Sudeep Salgia - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$