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Karan tells truth with probability 1/3 and lies with probability 2/3. Independently, Arjun tells truth with probability 3/4 and lies with probability 1/4. Both watch a cricket match. Arjun tells you that India won, Karan tells you that India lost. What probability will you assign to India's win?

  1. 1/2
  2. 2/3
  3. 3/4
  4. 5/6
  5. 6/7
asked in Probability by Veteran (38.7k points)   | 704 views

5 Answers

+3 votes
Best answer
Probability of India win =1/2

Probability of India lost=1/2

If really India wins, then Karan lies i.e.=2/3 and Arjun tells truth=3/4

Now prob. of Karan lies  and Arjun tells truth=2/3 * 3/4=1/2

 Now prob. of Arjun lies  and Karan tells truth=1/4 * 1/3=1/12

so, by Bayes theorem      1/2*1/2

                                 ------------------------------   =6/7

                                   1/2*1/2+1/2*1/12

so answer is (e)

PS: Assuming superover in case of tie.
answered by Veteran (58.4k points)  
selected by
what about match tie and both lies.?
I'm confused too according to official answer key this is the correct answer but it doesn't involve the case of a tie.

what to do if similar question is asked in gate?
+4 votes

Another Wrong Approach: (Did not considered the possibility of "TIE or DRAW or Any other event that can not decide a winner")

6/7 should be the correct answer.

Consider two events W & X:

W: India wins.

X : Arjun says India has won & Karan says India has lost.

We have to find the probability of India’s win, given that Arjun says India has won & Karan says India has lost.

i.e. We have to find P(W | X).

Now there are are two cases:

  1. India wins & Arjun says India has won and Karan says India has lost i.e. P(X | W)or
  2. India losses & Arjun says India has won and Karan says India has lost. P(X | ~W)

P(X | W) = Given that India has won, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun says truth and Karan lies.

= (3/4) x (2/3) = (1/2)

P(X | ~W) = Given that India has lost, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun lies & Karan says truth.

= (1 / 4) x (1 / 3) = (1/12)

Using Bayes’s theorem,

P(W | X) = P(X | W) / {P(X | W) + P(X | ~W)}

= (1 / 2) / {(1/2) + (1/12)}

= (6/12)/(7/12)

= 6/7.

Hence the probability of India’s win, given that Arjun says India has won & Karan says India has lost is 6/7.

answered by Veteran (13.1k points)  
+3 votes

2/3 must be the correct answer.

Consider four events W, L, T, X

W: India wins.

L : India loses.

T: Match ties.

X : Arjun says India has won & Karan says India has lost.

We have to find the probability of India’s win, given that Arjun says India has won & Karan says India has lost.

i.e. We have to find P(W | X).

Now there are three cases:

  1. India wins & Arjun says India has won and Karan says India has lost i.e. P(X | W) or
  2. India loses & Arjun says India has won and Karan says India has lost. P(X | L) or
  3. Match Ties & Arjun says India has won and Karan says India has lost. P(X | T) or

P(X | W) = Given that India has won, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun says truth and Karan lies.

= (3/4) x (2/3) = (1/2)

P(X | L) = Given that India has lost, what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun lies & Karan says truth.

= (1 / 4) x (1 / 3) = (1/12)

P(X | T) = Given that match has been tied , what is the probability that Arjun says India has won & Karan says India has lost.

= the probability that Arjun lies & Karan lies.

= (1 / 4) x (2 / 3) = (2/12)

Using Bayes’s theorem,

P(W | X) = P(X | W) / {P(X | W) + P(X | L) + P(X | T)}

= (1 / 2) / {(1/2) + (1/12) + (2/12)}

= (6/12)/(9/12)

= 6/9 = 2/3.

Hence the probability of India’s win, given that Arjun says India has won & Karan says India has lost is 2/3.

answered by Veteran (13.1k points)  
Not necessarily every match will end up with India's Win or Loss.It may end up without any result in case of tie, draw, suspension etc. We should consider those cases too & at that time both of them will lie.
yes.. You can see other answers. It was selected as a mistake. Thanks for notifying..
P(KaranTr)=1/3
P(KaranLi)=2/3

P(ArjunTr)=3/4
P(ArjunLi)=1/4

P(India win) = P(Win ArjunTr)*P(Win KaranLi) / [ P(Win ArjunTr)*P(Win KaranLi) + P(Loss ArjunLi)*P(Loss KaranTr) ]
             = (3/4)(2/3) / [(3/4)(2/3) + (1/4)(1/3)]
             = (1/2)(1/1) / [(1/2)(1/1) + (1/4)(1/3)]
             = 1/2 / 6/12+1/12 = 1 / 7/6 = 6/7
0 votes

WRONG APPROACH

The answer should be 1/2 i.e option A

Arjun told: India won,

Karan told: India lost,

Probability of India won = Probability that Arjun told truth(=3/4) & Karan lied(= 2/3)

So probability that India won = (3/4)x(2/3) = 1/2.

answered by Veteran (13.1k points)  
Is multiplication the correct thing to do here?
Yes sir! Since it is given in the question that truth telling of Karan & Arjun are independent events, so we can multiply them.

P(Arjun told truth ∩ Karan lied) = P(Arjun told truth) x P(Karan lied).
Suppose there are 10 people instead of 2. Now, with each of them telling truth or lie, the probability of win will decrease considerably rt? Or in the same way if you calculate P(loss) it won't be 1-P(win) rt?

Sir, here I have to assign the probability of Winning & losing.

Since I have not watched the match, anything that I am going to decide about India’s win or loss will be solely based on the reports I got from various people who have watched the match.

Here also P(Win) + P(Loss) need not be equal to 1, since there can be logical inconsistencies on the reports I got.

In this case for example:

There are 4 possible reports:

  1. Arjun Told Truth & Karan Told Truth,
  2. Arjun Told Truth & Karan Lied,
  3. Arjun Lied & Karan Told Truth,
  4. Both of them lied.

out of these 4 reports 2 are logically inconsistent in which both of them told truth or both of them lied.


Why 1 & 4 are logically inconsistent?? because Arjun told India won & Karan told India lost,

at the same time since India can’t loose AND win, Thus at the same time not both of them can tell truth or lie.

So points 1 & 4 are not satisfiable & hence they are invalid/inconsistent.

So Actually P(Win) + P(Loss) + P(Inconsistencies) = 1.


For example here

P(win) = P(report 2) = ½

P(Loss) = P(report 3) = 1/12

P(Inconsistent report) = P(report 1) + P(report 4) = 5/12.

It can be observed that P(Inconsistencies) is eating up a notable fraction of our total probability.

As number of reports will increase probability of inconsistencies will increase so OUR DEDUCTIONS about P(Win) & P(Loss) may get affected, but since those reporters are not going to decide India’s win or loss so ACTUAL RESULT is not going to be affected..

They will only affect our deductions about India’s win or loss. right sir?

Nopes. You are wrong. But I don't know how to explain it- P(Inconsistencies) must be 0. It is application of conditional probability.
Yes sir, I might wrong, but presently I am unable to figure out my mistake here.

I solved similar types of questions previously, & applied conditional probability there to get the correct answers.
what is answer here @Arjun ?

I feel like it should be 6/7. Please answer this one !
0 votes
Case 1 : Arjun tells the truth that India won , then Karan  must lie about the fact India lost , in order India to win .

Therefore :   (3/4)*(2/3) = (1/2)

Case 2: Arjun is lieing about India's win then karan must also lie about India's loss because the motive is to make India win .

Therefore (1/4)*(2/3) = 1/6 .

Total probability for India wining is 1/2 + 1/6 = 2/3 .

Hence answer is B .
answered by Boss (7.1k points)  
reshown by

This approach is not correct. Other approaches are given here :)

http://gateoverflow.in/18499/tifr2010-a-19

This question seems like completely repeated .. Not even names are changed !


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