Answer
$P(A) = 0.5$
Work Step by Step
Given
$P(A) + P(B) = 0.9\quad\quad\quad(1)$
$P(A|B) = 0.5$
$P(B|A) = 0.4$.
We have to find $P(A)$.
$P(A|B) = \frac{P(A \cap B)}{P(B)}\quad\quad\quad(2)$
$P(B|A) = \frac{P(A \cap B)}{P(A)}\quad\quad\quad(3)$
Using $(2)$ and $(3)$ we get that
$\frac{P(A)}{P(B)} = \frac{P(A|B)}{P(B|A)} = \frac{0.5}{0.4}=\frac{5}{4}$
=>$P(B) = \frac{4P(A)}{5} = 0.8P(A)$
Putting the value in eq. $(1)$ we get,
$P(A) + 0.8P(A) = 0.9$
$=> P(A) = \frac{0.9}{1.8} = \frac{1}{2} = 0.5$
