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.