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I have been trying this question :-

Amar and Akbar both tell the truth with probability 3/4 and lie with probability 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?

And I came up with this solution which i have shown below :-

India Won(Actually) ------ 4 cases (i) Amar Truth And Akbar Truth (ii) Amar Truth And Akbar False (iii) Amar False And Akbar Truth (iv) Amar False And Akbar False

India Lost(Actually) ----- 4 cases (i) Amar Truth And Akbar Truth (ii)Amar Truth And Akbar False (iii)Amar False And Akbar Truth (iv)Amar False And Akbar False

Now let e1 : India win ; e2 : Akbar told that "Amar told me that India won"

P(e1|e2) = P(e1 AND e2) / P(e2)
         = 
         (P("Amar Truth And Akbar Truth") 
       + P("Amar False And Akbar False"))/(P(Amar Truth And Akbar False) 
       + P(Amar False And Akbar Truth) 
       + P("Amar Truth And Akbar Truth") 
       + P("Amar False And Akbar False"))

   P(e1|e2) = (10/16)/(16/16) ===> 10/16 ----> Answer

I dont see any mistake in my logic. But when i look at others logic on the internet, they say that the problem cant be solved because P(India win) is not given in the question. But I think P(India win) is not required here.

Please help me out in this. Thanks in advance.

Sanku
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    Take a closer look at your denominator. Your denominator should reflect the probability that anthony is told that akbar is told by amar that india won. The probability of this happening is not $1$. – JMoravitz Mar 19 '18 at 02:02
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    Denoting that Amar told the truth as $M_t$ and lied as $M_f$, and similarly that Akbar told the truth as $K_t$ and lied as $K_f$, and India actually winning as $I_w$ and losing as $I_\ell$, the denominator should have read as $P(M_t\cap K_t\cap I_w)+P(M_t\cap K_f\cap I_\ell)+P(M_f\cap K_t\cap I_\ell)+P(M_f\cap K_f\cap I_w)$. You neglected to consider whether or not India won or not in your calculations in the denominator. Indeed, there is in fact not enough information to simplify the denominator and the final answer varies based on the value of $P(I_w)$. – JMoravitz Mar 19 '18 at 02:09
  • Ohh okay got it thanks :). One more doubt. We can't just assume that probabilities of india winning and losing as 0.5 and 0.5 and solve the question right? – Sanku Mar 19 '18 at 02:27
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    No you may not. I would argue that you shouldn't even be allowed to assume that Amar and Akbar's decisions to tell the truth or lie are independent of eachother or of the match. In an extreme example, maybe india wins with probability $\frac{1}{4}$, Amar always says that India lost, and Akbar always says that Amar told him that India won and Anthony knows all this except the probability India wins. In this hypothetical situation all of the explicitly written hypotheses are met, they each lie $\frac{1}{4}$ of the time, but Anthony does not receive any information that influences his knowledge. – JMoravitz Mar 19 '18 at 03:35
  • yeah I think I got it now. Thank you for the example. – Sanku Mar 19 '18 at 06:36
  • I am trying to make my concept clear. So, lets assume that "Arjun and Amar telling lie or truth are independent of each other or the outcome of the match".. Now, take another example that, Arjun told me that India has won the match and Amar told me India has lost the match. So, now we don't need the probability of India winning right. Since I can use Bayes theorem To calculate my answer P(India win |e2). numerator will be product of P(India win|arjun truth and amar false) and denominator will be P(India win|arjun truth and amar false) + P(India lose|arjun false and amar truth). – Sanku Mar 19 '18 at 06:56
  • Does this answer your question? What probability would you assign to India's win? – AKP2002 Dec 23 '21 at 10:35

0 Answers0