India's win

Let $W$ be the event that India wins, (and $\overline{W}$ be the event that it loses), and $A$ be the event that Arjun says that India has won and Karan says that India has lost. Then, assuming that the match can't end in a tie,

$P(W | A) = \dfrac{P(W \cap A)}{P(A)} = \dfrac{P(W \cap A)}{P(W \cap A) + P(\overline{W} \cap A)}.$

Now $P(W \cap A)$ is the probability that Arjun is telling the truth and Karan is lying, which comes out to $\dfrac{3}{4} * \dfrac{2}{3} = \dfrac{1}{2}.$

Next, $P(\overline{W} \cap A)$ is the probability that Arjun is lying and Karan is telling the truth, which comes out to $\dfrac{1}{4} * \dfrac{1}{3} = \dfrac{1}{12}.$

Thus $P(W | A) = \dfrac{\dfrac{1}{2}}{\dfrac{1}{2} + \dfrac{1}{12}} = \dfrac{\dfrac{6}{12}}{\dfrac{7}{12}} = \boxed{\dfrac{6}{7}}.$