Karan tells truth with probability 1/3 and lies with probability 2/3. Independently, Arjun tells truth with probability 3/4 and lies with probability 1/4. Both watch a cricket match. Arjun tells you that India won, Karan tells you that India lost. What probability will you assign to India’s win?
Probability India won = 1/2
Prob for India lost = 1/2
If definitely, India won : then
Arjun is telling the truth and karan is telling a lie then
arjun truth and karan lies = 3/4*2/3 = 1/2
arjun lies and karan truth = 1/4*1/3 = 1/2
Apply bayes therome now (1/2*1/2) /(1/2*1/2 + 1/2*1/12) = 6/7 is the answer
There is a twist here actually. If the match is already over then you will assign probability of 1/2 to india’s win irrespective of the statements given by arjun and karan.
More appropriate question would have been, “arjun tells you that india will win, karan tells you that india will loose”.
In the above case we can go for bayes theorem and answer would be 6/7 as proposed by mukul_d.