$A$ die is thrown three times. Events $A$ and $B$ are defined as below:
$A$: $4$ on the third throw
$B$: $6$ on the first and $5$ on the second throw
Find the probability of $A$ given that $B$ has already occurred.

Given $P(A)=0.5, P(B)=0.4, P(A \cap B)=0.3$,then $P(A^{\prime} / B^{\prime})$ is equal to

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 $B$)

Let $A$ and $B$ be independent events with $P(B) = \frac{2}{5}$ and $P(A \cup B) = \frac{11}{20}$. Then $P(A' \mid B)$ is a root of which equation?

Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then,which of the following is $TRUE$?

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