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20 votes
20 votes

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
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4 Comments

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.
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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??

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Yes. Mostly in any question about win/loss tie is ignored. If game is cricket whichever team hits more boundaries will be the winner 😉
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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.😶
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11 Answers

14 votes
14 votes
Best answer

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


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

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

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

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


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

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

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

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


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

edited by

4 Comments

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

..............

by

3 Comments

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
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Akriti sood because in those cases India is not winning

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

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. 

1 comment

Nice explanation.
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16 votes
16 votes

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


https://www.youtube.com/watch?v=yBn-eGrRHJU


3 Comments

Thank you very much. Nice Explanation.
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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

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Indeed a great explantion.

 

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

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