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

$A$ couple has two children. Find the probability that both children are females,if it is known that the elder child is a female.

If $P(A) = \frac{6}{11}$,$P(B) = \frac{5}{11}$ and $P(A \cup B) = \frac{7}{11}$,find $P(A | B)$.

An unbiased die is thrown twice. Let the event $A$ be 'odd number on the first throw' and $B$ the event 'odd number on the second throw'. Check the independence of the events $A$ and $B$.

Two events $A$ and $B$ are such that $P(A)=\frac{1}{4}$, $P(A|B)=\frac{1}{4}$ and $P(B|A)=\frac{1}{2}$. Consider the following statements:
$(I) P(\bar{A}|\bar{B})=\frac{3}{4}$
$(II) A$ and $B$ are mutually exclusive
$(III) P(A|B)+P(A|\bar{B})=1$
Then,

Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability $0.7$ whereas George,independently,hits the target with probability $0.4$. Given that exactly one shot hit the target,what is the probability that it was George's shot?