If $P(A) = \frac{3}{10}$,$P(B) = \frac{3}{5}$,and $P(A \cup B) = \frac{3}{5}$,then find the value of $P(A|B) \times P(B|A)$.

If $A$ and $B$ are two independent events,then $A$ and $\bar{B}$ are

$A$ and $B$ are two groups of books. Group $A$ consists of $8$ science and $5$ engineering books,and group $B$ consists of $6$ science and $7$ engineering books. When an unbiased die is rolled,if $2$ or $5$ turns up,a book is selected at random from group $A$. Otherwise,a book is selected at random from group $B$. The probability of selecting a science book is

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 $TRUE$?

Two integers are selected at random from the set $\{1, 2, \dots, 11\}.$ Given that the sum of the selected numbers is even,what is the conditional probability that both numbers are even?

If $A$ and $B$ are two events of a sample space $S$ such that $P(A)=0.2$,$P(B)=0.6$ and $P(A \mid B)=0.5$,then $P(A^{\prime} \mid B) = $