Thank you for reporting, we will resolve it shortly
Q.
If $P ( A / B )+ P (\overline{ A } / \overline{ B })=1$ and $0< P ( A )<1$; and $0< P ( B )<1$ the which of the following does not hold good?
Probability - Part 2
Solution:
$\frac{ P ( A \cap B )}{ P ( B )}+\frac{ P (\overline{ A } \cap \overline{ B })}{ P (\overline{ B })}=1$
$\frac{P(A \cap B)}{P(B)}=1-\frac{1-P(A \cup B)}{1-P(B)}=\frac{1-P(B)-1+P(A \cup B)}{1-P(B)} $
$\frac{P(A \cap B)}{P(B)}=\frac{P(A \cup B)-P(B)}{1-P(B)}=\frac{P(A)-P(A \cap B)}{1-P(B)} $
$\therefore P(A \cap B)-P(A \cap B) P(B)=P(A) P(B)-P(B) P(A \cap B) $
$ P(A \cap B)=P(A) P(B) \Rightarrow (C)$