If $P \left(A\right) = 0.4$, $P \left(B\right) = 0.7$ and $P \left(B | A\right) = 0.5$, find $P \left(A | B\right)$ and $P \left(A \cup B\right)$
By definition, $P \left(B | A\right) = \dfrac{P \left(A \cap B\right)}{P \left(A\right)}$
i.e. $\;$ $0.5 = \dfrac{P \left(A \cap B\right)}{0.4}$
$\implies$ $P \left(A \cap B\right) = 0.5 \times 0.4 = 0.2$
Now, $P \left(A | B\right) = \dfrac{P \left(A \cap B\right)}{P \left(B\right)}$
$\implies$ $P \left(A | B\right) = \dfrac{0.2}{0.7} = \dfrac{2}{7}$
By definition, $P \left(A \cup B\right) = P \left(A\right) + P \left(B\right) - P \left(A \cap B\right)$
i.e. $\;$ $P \left(A \cup B\right) = 0.4 + 0.7 - 0.2 = 0.9 = \dfrac{9}{10}$