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 $P\left(\frac{B}{A}\right)$.

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails } | \text{ } B \text{ has failed})$.

If $A$ and $B$ are two events such that $P(A)=\frac{1}{2}$,$P(B)=\frac{1}{2}$ and $P(A \mid B)=\frac{1}{4}$,then $P(A^{\prime} \cap B^{\prime})$ is

$A$ and $B$ are independent events of a random experiment if and only if

An urn contains $5$ red and $2$ green balls. $A$ ball is drawn at random from the urn. If the drawn ball is green,then a red ball is added to the urn and if the drawn ball is red,then a green ball is added to the urn; the original ball is not returned to the urn. Now,a second ball is drawn at random from it. The probability that the second ball is red,is

$A$ college student has to appear for two examinations $A$ and $B$. The probabilities that the student passes in $A$ and $B$ are $\frac{2}{3}$ and $\frac{3}{4}$ respectively. If it is known that the student passes at least one among the two examinations,then the probability that the student will pass both the examinations is