Let $A$ and $B$ be not mutually exclusive events. If $P(A) = \frac{4}{9}$ and $P(A \cap \bar{B}) = \frac{3}{7}$, then find $P\left(\frac{B}{A}\right)$.

If $A$ and $B$ are mutually exclusive events with $P(B) \neq 1$, then $P(A \mid \bar{B})$ is equal to (Here $\bar{B}$ is the complement of the event $B$)

If $A$ and $B$ are two events such that $P(A) = \frac{1}{3}$,$P(B) = \frac{1}{5}$,and $P(A \cup B) = \frac{1}{3}$,then the value of $P(A^{\prime} | B^{\prime}) + P(B^{\prime} | A^{\prime})$ is:

$A$ fair die is rolled. Consider events $E=\{1,3,5\}, F=\{2,3\},$ and $G=\{2,3,4,5\}$. Find $P((E \cup F) | G)$ and $P((E \cap F) | G)$.

One card is drawn at random from a well-shuffled deck of $52$ cards. In which of the following cases are the events $E$ and $F$ independent?
$E:$ 'the card drawn is black'
$F:$ 'the card drawn is a king'

Let $E_{1}$ and $E_{2}$ be two events such that the conditional probabilities $P(E_{1} \mid E_{2}) = \frac{1}{2}$,$P(E_{2} \mid E_{1}) = \frac{3}{4}$ and $P(E_{1} \cap E_{2}) = \frac{1}{8}$. Then: