In the question on probability of India winning, why did we assume Aman and Bala always make contradictory statements? Why didnt we add probability of India winning if they both make same statements where they both are either wrong or right?

Welcome @writetovamshi.

Because in the question it is mentioned that they are giving contradictory statements.

The final question statement was

Bala tells you India won the match, whereas Aman tells you that India had lost.

This means that only one of them is telling the truth.

I hope it clears your doubt.

@hemant_patel : Thanks for the clarity. I thought it was an example of one of their statements and not the only statement they make.

I’m glad that i cleared that up for you.

I’m far more confused than the others at this question, how did we calculate the probability of India Winning like this, also, it says they made independent decisions, i.e. if one person tells the truth it doesn’t imply other will tell lie. Can anybody give an elaborate explanation to this question like we generally find at stack exchange.

I’d really appreciate the effort

I also don’t understand the Baye’s theorem implication in this question, it totally seem weird to me. Please help!

I accepted whatever answer @hemant_patel gave . They make only 1 statement. But then, a lot more doubts arose after that. If they are making only statement, then their probabilities are same as India winning probability. There is lot of ambiguity in either the question or the answer, depending on what the professor has in mind.

No it’s not necessary that if one is telling the lie other will tell lie too. But in the question it is given that Bala and Aman is telling different things.

The Baye’s theorem is applied because we want to update probability of winning. Earlier the winning probability is 0.5 but after the circumstances happened in the question, the probability will change because now it is depending on some other events too. In this case it is depending on Aman or Bala telling some news.

can you send me a detailed solution to this if you do have one, Idk why but its just not getting through me. This is the only question I got stuck at

I will try to provide one by tomorrow.

Hi. I am also having trouble understanding the solution to this question. Can you please put up a detailed solution here?

See if rephrasing the solution helps:

Given: Bala(B) tells India won, while Aman(A) tells India lost.

Since *the statements made by A and B are contradictory (key point)*, it means one of them is telling truth and other is lying (*only conclusion possible from above*).

Total probability of such a scenario(making contradictory statements) will be:

P(A and B makes contradictory statement) = P(A\bar{B}) + P(\bar{A}B) = \frac{1}{3} * \frac{1}{4} + \frac{2}{3} * \frac{3}{4} = 7/12

For India to really win, i.e. favourable case is Aman lying(that India lost) and Bala telling truth(that India win) .

P(Favourable case for India winning) = P(\bar{A}B) = \frac{2}{3} * \frac{3}{4} = 1/2

Finally,

P(India wins given A and B make contradictory statements) \\= \frac{P(Favourable\ Case)}{P(Total\ Cases)} = \frac{P(\bar{A}B)}{P(A\bar{B})+P(\bar{A}B)} = \frac{1/2}{7/12} = 6/7