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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
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11 Answers

13 votes
13 votes

so D is the ans

11 votes
11 votes

Tree-Diagram in such problems make them easier to solve

Let's say India Won.

In Two ways probability can be assigned to India's Win

The two favorable cases of our problem are encircled by green.

Case 1: Both Amar and Akbar Tell Truth

Actual Case: India Won

Amar Tell Akbar: India Won

Akbar Tells Anthony : Amar Told me India Won (Yes this is true)

So probability for this = $\frac{3}{4} * \frac{3}{4} = \frac{9}{16}$

Case 2: Amar Tells lies and Akbar also Lies

Actual Case: India Won

Amar Told Akbar: India Lost

Akbar Told Anthony: Amar told me India Won 

So, Reversing statement of Akbar we get "Amar told me India lost"

This means Amar Told "India has lost" to Akbar

Since Amar also lied, so This means India Must have Won!!.

So, Probability for this case = $\frac{1}{4} * \frac{1}{4} = \frac{1}{16}$

Total Probability = $\frac{10}{16}$

4 votes
4 votes

I feel every answer which is given here is wrong (in a way). I know even the official response is D. (10/16). However, Correct answer should be e. None of these. Actually this probability CAN NOT BE CALCULATED AT ALL.  First let me explain my answer and then I shall explain where everyone (yeah even the official answer key) is making mistake.

First let me introduce some notations:

Notations:

      Amar telling truth will be denoted as event A.           ∴ Amar telling lie is denoted by Ac.

      Akbar telling truth will be denoted as event B.          ∴ Akbar telling lie is denoted by Bc.

      India actually winning will be denoted as event W.   ∴  India actually losing will be denoted as event Wc.

Now, Lets talk about the sample space. There are 8 members inside the sample space.

1. (India Won, Amar told truth, Akbar told truth) denoted as (W, A, B)

2. (India Won, Amar told truth, Akbar told lie) denoted as (W, A, Bc)

3. (India Won, Amar told lie, Akbar told truth)  denoted as (W, Ac, B)

4. (India Won, Amar told lie, Akbar told lie) denoted as (W, Ac, Bc)

5. (India lost, Amar told truth, Akbar told truth) denoted as (Wc, A, B)

6. (India lost, Amar told truth, Akbar told lie) denoted as (Wc, A, Bc)

7. (India lost, Amar told lie, Akbar told truth) denoted as (Wc, Ac, B)

8. (India lost, Amar told lie, Akbar told lie) denoted as (Wc, Ac, Bc)

Let me explain the members of the sample space.

Consider the first member (India Won, Amar told truth, Akbar told truth). What I mean to say is that India actually won the match and then Amar told Akbar that India won the match(truth) and finally Akbar told Anthony that India won the match(truth).

Similarly consider seventh member: (India lost, Amar told lie, Akbar told truth). What this means is that India actually lost the match and then Amar told Akbar that India won the match(lie) and finally Akbar told Anthony that India won the match(truth). Note that Akbar's truth is not on the basis of absolute truth. Akbar's truth is on the basis of Anthony's information only. So here Akbar telling truth means whatever he heard from Anthony, he just says exactly the same thing. And Akbar telling lie is that whatever he heard from Anthony, he tells completely opposite things.


A basic concept: If it is said that given an event A has occurred, what is the probability of B occurring, then it is given by the expression: P (B | A)


Now onto the question:
The question is asking to calculate that given that Akbar told Anthony that India Won, what is the probability of India's actually winning ?
i.e. the problem is asking P(W | Akbar told Anthony that India Won)  = ?. Note that the problem does not ask us to find P(W).

Now let us see from our sample space, what are the possible ways in which Akbar can tell that India has won:

1. (W, A, B) .   i.e. India can actually win, Amar tells truth (that India won), Akbar tells truth(that India won) .

2. (W, Ac, Bc) .  i.e. India can actually win, Amar tells lie (that India lost), Akbar tells lie (that India won) .

3. (L, A, Bc) .  i.e. India can actually lose, Amar tells truth (that India lost), Akbar tells lie (that India won) .

4. (L, Ac, B) .  i.e. India can actually lose, Amar tells lie (that India won), Akbar tells truth (that India won) .

These 4 are the only possible ways in which Akbar can tell that India has won.

Now the favorable cases are the ones where India has actually won. Thus the favorable cases are:

1. (W, A, B)

2. (W, Ac, Bc)

Thus the problem boils down to :

$P(W | \text{ Akbar told Antony that India Won}) = \LARGE\frac{P(W, A, B)+P(W, A^{c}, B^{c})}{P(W, A, B)+P(W, A^{c}, B^{c})+P(W^{c}, A^{c}, B)+P(W^{c},A, B^{c}))}$

Now we need to calculate the individual probabilities.

P(W, A, B) = P(W) * P(A | W) * P(B | W ∩ A). Now probability of Amar/Akbar telling truth/lie is independent of anything else. Thus P(A|W) = P(A) and P(B | W ∩ A) = P(B) ∴ P(W, A, B) = P(W) * P(A) * P(B)

Likewise P(W, Ac, Bc)=P(W) * P(Ac) * P(Bc) .  Similarly for the other probabilities.

We know P(A)=P(B)=3/4 and P(Ac)=P(Bc)=1/4. However we dont know P(W) or P(Wc).

So let us assume P(W) = p ∴ P(Wc) = 1-p and hope that while evaluating the above expression all p terms get cancelled.

$\LARGE \frac{P(W, A, B)+P(W, A^{c}, B^{c})}{P(W, A, B)+P(W, A^{c}, B^{c})+P(L, A^{c}, B)+P(L,A, B^{c}))} \\ \LARGE = \frac{p\frac{3}{4}\frac{3}{4} + p\frac{1}{4}\frac{1}{4}}{p\frac{3}{4}\frac{3}{4} + p\frac{1}{4}\frac{1}{4}+(1-p)\frac{1}{4}\frac{3}{4}+(1-p)\frac{3}{4}\frac{1}{4}}\\ \LARGE =\frac{\frac{10}{16}p}{\frac{10}{16}p-\frac{6}{16}p+\frac{6}{16}} =\frac{10p}{4p+6}$

We see that it is dependent on the actual probability of India Winning(p) which is not given anywhere in the question. Thus the answer cannot be found at all.

Interestingly enough, if we assume p=0.5 (Chances of India winning and losing are same), we get that

$\LARGE\frac{10*0.5}{4*0.5+6}=\frac{5}{8}=\frac{10}{16} !!!$ But there is no reason why p should be 0.5. Thus Option (d) 10/16 is NOT the answer.

Now let me explain where other answers have made mistakes:

1. by Anurag Pandey:

He mistakenly assumed the following equation:

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

The correct Bayes Theorem equation should have been:

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

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0 votes
0 votes

All the answers given here are wrong in a way. The question given is not solvable because "the probability of india winning " is not given in the question

Let e1 : India wins ; e2 : Akbar says that "Amar told me that India won"

now in the Numerator(e1 "AND " e2) there will be two cases

case (i) :- India actually winning and both amar and akbar telling the truth (i.e. IndiaWin AND AmarTruth AND AkbarTruth)

case (ii) :- India actually winning and both amar and akbar telling lies.       (i.e. IndiaWin AND AmarFalse AND AkbarFalse)

Denominator will be the reduced sample space of event e2 i.e. ("Akbar telling anthony that ''Amar told him that india won'").

Here we need to consider the cases where india winning and also india losing which will give us 4 cases :

case (i) :- India actually winning and both amar and akbar telling the truth (i.e. IndiaWin AND AmarTruth AND AkbarTruth)

case (ii) :- India actually winning and both amar and akbar telling lies.       (i.e. IndiaWin AND AmarFalse AND AkbarFalse)

case (iii) :- India actually losing and amar telling truth and akbar telling lie (i.e. IndiaLose AND AmarTruth AND AkbarFalse)

case (iv) :- India actually losing and amar telling lie and akbar telling truth (i.e. IndiaLose AND AmarFalse AND AkbarTruth)

Now writing all of them in equation, we get one which I have written in the image

But to find the solution, we need P(India winning) is not given, so no sufficient data to solve it.  

Note :- Everyone are considering only the cases where india wins. But we need to consider the cases where the india loses also, since the sample space of event e2 contains both india winning and also india losing.

I am attaching also the tree diagram which may make it easier to understand.

Also we can't just assume that "arjun and amar telling lies are independent of the outcome of the match or not". So, in either way, this question is unsolvable. There's just no sufficient information for anthony to come to a conclusion.

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