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Karan tells truth with probability $\dfrac{1}{3}$ and lies with probability $\dfrac{2}{3}.$ Independently, Arjun tells truth with probability $\dfrac{3}{4}$ and lies with probability $\dfrac{1}{4}.$ 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. $\left(\dfrac{1}{2}\right)$
2. $\left(\dfrac{2}{3}\right)$
3. $\left(\dfrac{3}{4}\right)$
4. $\left(\dfrac{5}{6}\right)$
5. $\left(\dfrac{6}{7}\right)$

edited | 1.9k views
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The answer depends on Absolute Probability of India's winning. If Win and Lose are only two options then 6/7 is the answer. If Win, Lose and Tie are the options then 2/3 is the answer. Required probability comes out to be 6p/(5p+1) wherer p is the absolute probability of India winning. What is the answer according to TIFR answer key?

If really India wins, then Karan lies  $\left(P= \frac{2}{3}\right)$ and Arjun tells truth $\left(P=\frac{3}{4}\right)$

Now probability of Karan lying and Arjun telling truth $=\dfrac{2}{3}\times \dfrac{3}{4}=\dfrac{1}{2}$

Now probability of Arjun lying and Karan telling truth $=\dfrac{1}{4} \times \dfrac{1}{3}=\dfrac{1}{12}$

So, by Bayes theorem,

Probability of India winning $=\dfrac{\dfrac{1}{2}}{\dfrac{1}{2} + \dfrac{1}{12}}=\dfrac{6}{7}$

So, answer is $(e)$

PS: Assuming superover in case of tie.
by Veteran (117k points)
edited by
+1
what about match tie and both lies.?
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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?
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PS: Assuming superover in case of tie.

@venky.victory35 @Pankaj Joshi and @srestha I think tie case will not come in picture because statement -->"Arjun tells you that India won, Karan tells you that India lost" will not remain valid (even Arjun tells T or F). Same could be observed via tree diagram.

Same thing is mentioned by  @अनुराग पाण्डेय

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

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@Sreshtha

What does the term in denominator imply?

I mean what is the English equivalent statement of the denominator term used?
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denominator is total probability of india win

numerator arjun tells india win and india won
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Arjun tells truth need not imply that karan tells false right ? Both could tell the truth right  and still India could win right ? also both could lie and India could win, then why in denominator those cases were not considered, i quote as you said above comment

"denominator is total probability of india win"

Please let me know if i misinterpreted what you meant

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@Arjun Sir, what is the sample space of this problem,so that I can choose the India Win part  & also calculate the total probability from that.

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@srestha why did you consider this part, Arjun said India won the match but if he is lying it means india didn't win the match then why did you consider it. 0

but if he is lying it means india didn't win the match

why r u considering this?

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@srestha i asked you question and you asked the same back to me :/
My question was why did you consider Arjun Lying and Karan telling truth in total probability because if Arjun lied it means that india didn't win the match and we are looking for the probability of India's win.

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ok got ur point

$\frac{\text{probability of wining by their statement}}{\text{probability of wining or losing by their statement}}$

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.

by Boss (14.1k points)

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.

by Boss (14.1k points)
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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.
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yes.. You can see other answers. It was selected as a mistake. Thanks for notifying..
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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
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I did not understand the denominator. @s.abhishek1992
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Anurag Pandey Sir as u have said "Arjun lies & Karan lies."  then match ties can u please tell what will happen if both saying truth.

 $\text{India Win}$ $\text{India Loss}$ $\text{Marginal}$ $\text{True Event}$ $\text{Arjun True } \cap \text{Karan Lies}$ $\text{Arjun Lies } \cap \text{Karan True}$ $\text{Sum of Row 1 Events}$ $\text{Flase Event}$ $\text{Arjun True} \cap \text{Karan True}$ $\text{Arjun False} \cap \text{Karan False}$ $\text{Sum of Row 2 Events}$ $\text{Marginal}$ $\text{Sum of Column 1 Events}$ $\text{Sum of Column 2 Events}$ $\text{Marginal Sum}$

 $\text{India Win}$ $\text{India Loss}$ $\text{Marginal}$ $\text{True Event}$ $\frac{3}{4} \times \frac{2}{3}$ $\frac{1}{4} \times \frac{1}{3}$ $\frac{7}{12}$ $\text{Flase Event}$ $\frac{3}{4} \times \frac{1}{3}$ $\frac{1}{4} \times \frac{2}{3}$ $\frac{5}{12}$ $\text{Marginal}$ $\frac{9}{12}$ $\frac{3}{12}$ $1$

Probability will assign to India's win = $P\text{(True Event/Indian Win)} = \frac{\frac{6}{12}}{\frac{7}{12}} = \frac{6}{7}$

by Boss (11.2k points)
edited

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.

by Boss (14.1k points)
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Is multiplication the correct thing to do here?
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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).
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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?
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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?

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Nopes. You are wrong. But I don't know how to explain it- P(Inconsistencies) must be 0. It is application of conditional probability.
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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.
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what is answer here @Arjun ?

I feel like it should be 6/7. Please answer this one !
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 .

by Loyal (7.4k points)
reshown
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This approach is not correct. Other approaches are given here :)

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

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