2/3 must be the correct answer.

Consider four events W, L, T, X

W: India wins.

L : India loses.

T: Match ties.

X : Arjun says India has won & Karan says India has lost.

We have to find the probability of India’s win, given that Arjun says India has won & Karan says India has lost.

i.e. We have to find P(W | X).

Now there are three cases:

- India wins & Arjun says India has won and Karan says India has lost i.e. P(X | W) or
- India loses & Arjun says India has won and Karan says India has lost. P(X | L) or
- Match Ties & Arjun says India has won and Karan says India has lost. P(X | T) or

P(X | W) = Given that India has won, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun says truth and Karan lies.

= (3/4) x (2/3) = (1/2)

P(X | L) = Given that India has lost, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun lies & Karan says truth.

= (1 / 4) x (1 / 3) = (1/12)

P(X | T) = Given that match has been tied , what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun lies & Karan lies.

= (1 / 4) x (2 / 3) = (2/12)

Using Bayes’s theorem,

P(W | X) = P(X | W) / {P(X | W) + P(X | L) + P(X | T)}

= (1 / 2) / {(1/2) + (1/12) + (2/12)}

= (6/12)/(9/12)

= 6/9 = 2/3.

Hence the probability of India’s win, given that Arjun says India has won & Karan says India has lost is 2/3.