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Amar and Akbar both tell the truth with probability $\dfrac{3 } {4}$ and lie with probability $\dfrac{1}{4}$. Amar watches a test match and talks to Akbar about the outcome. Akbar, in turn, tells Anthony, "Amar told me that India won". What probability should Anthony assign to India's win?

1. $\left(\dfrac{9}{16}\right)$
2. $\left(\dfrac{6 }{16}\right)$
3. $\left(\dfrac{7}{16}\right)$
4. $\left(\dfrac{10}{16}\right)$
5. None of the above

Looks like an ambiguous question.  Antony can assign probability based on both actuality and Akbars' information. Later one is answered in this thread. Former one seems to make it 1/2.

I just want to know that in this ques and in this https://gateoverflow.in/18499/tifr2010-a-19-tifr2014-a-6 ,we would not consider the case of tie unless specified??

Yes. Mostly in any question about win/loss tie is ignored. If game is cricket whichever team hits more boundaries will be the winner 😉
I don’t know the probability to the Event asked in the Question. But is know that the Probabillity that you will go mad solving these question model is 1.😶

Option D should be the correct answer.

Consider the following events,

$W$ : India wins,

$W\neg$ : India does not wins (India Lost/ Match Draw/ Match Tie/ Match Suspended etc.)

$X$ : Akbar tells Anthony, "Amar told me that India won"

$X\neg$ : Akbar tells Anthony, "Amar told me that India did not won"

Given $X$, we have to find $W$, that is we have to calculate $P\left(\frac{W}{X}\right)$.

$P\left(\frac{W}{X}\right)$ can be calculated using Bayes's theorem as follow:

$P\left(\dfrac{\text{India Wins}}{\text{Akbar tells Anthony “Amar told me that India won"}}\right)$

$= \dfrac{P\left(\dfrac{\text{Akbar tells Anthony “Amar told me that India won"}}{\text{India Wins}}\right)}{P\left(\frac{\text{Akbar tells Anthony “Amar told me that India won"}}{{\text{India Won}}}\right)\cup P\left(\frac{\text{Akbar tells Anthony “Amar told me that India won"}}{\text{India didn't won}}\right)}$

rewriting same equation using the events defined:

$P\left(\frac{W}{X}\right) = \frac{P\left(\frac{X}{W}\right)}{P\left(\frac{X}{W}\right) + P\left(\frac{X}{W\neg}\right)}\\$

Calculation of $P\left(\frac{X}{W}\right) and \ P\left(\frac{X}{W\neg}\right)\\$ :

$P\left(\frac{X}{W}\right) = P\left(\frac{Case \ 1}{W}\right) \cup \ P\left(\frac{Case \ 4}{W}\right)\\$

$P\left(\frac{Case \ 1}{W}\right) = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}\\$

$P\left(\frac{Case \ 4}{W}\right) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\\$

$So \ P\left(\frac{X}{W}\right) = \frac{9}{16} + \frac{1}{16} = \frac{10}{16}\\$

$P\left(\frac{X}{W\neg}\right) = P\left(\frac{Case \ 6}{W\neg}\right) \cup \ P\left(\frac{Case \ 7}{W\neg}\right)\\$

$P\left(\frac{Case \ 6}{W\neg}\right) = \frac{3}{4} \times \frac{1}{4} = \frac{3}{16}\\$

$P\left(\frac{Case \ 7}{W\neg}\right) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}\\$

$So \ P\left(\frac{X}{W\neg}\right) = \frac{3}{16} + \frac{3}{16} = \frac{6}{16}\\$

$Hence, \ P\left(\frac{W}{X}\right) = \frac{\frac{10}{16}}{\frac{10}{16} + \frac{6}{16}} = \frac{10}{16}.\\$

yup i hav done it using tree diagram  ...r u able to understand wat r the cases according to answer ???
No. I think these cases made solution more complicated
Hi, your denominator is summing upto 1 which will never be the case.

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why are not considering the other cases??when amar says truth and albar lies
when amar lies and akbar says truth.??
pls explain.i get confused in these questions

Akriti sood because in those cases India is not winning

You need to consider the case where India is not winning also. Check my solution

there are two cases in which india win:
1) Akbar tells the truth and Amar tells the truth : 3/4*3/4 = 9/16
2) Akbar tells a lie that India loose and Amar tells lie to anthony that "Akbar told me india win" : 1/4*1/4 = 1/16

So total probability of winning India would be 9/16 + 1/16 = 10/16

Source : http://www.careercup.com/question?id=13438685

I felt this is easy to understand.

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### 1 comment

Nice explanation.

watch this as well for the better understanding of the solution :

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Thank you very much. Nice Explanation.

Question ask out of total winning message Anthony gets what all are true.

P(win/winMsg) = (WTrTr) + (WLiLi)  / [ (WTrTr) + (WLiLi) + (WcTrLi) + (WcLiTr) ]

P(win/winMsg) = (3/4)2 + (1/4)2 / [(3/4)2 + (1/4)2  + (3/4)(1/4) +(3/4) (1/4)]=10/16

Indeed a great explantion.

Adding some point in the above the favourable cases are those which are satisfying “amar told me that india won”.