TIFR CSE 2010 | Part A | Question: 19, TIFR CSE 2014 | Part A | Question: 6

Question

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?

• A. $\left(\dfrac{1}{2}\right)$
• B. $\left(\dfrac{2}{3}\right)$
• C. $\left(\dfrac{3}{4}\right)$
• D. $\left(\dfrac{5}{6}\right)$
• E. $\left(\dfrac{6}{7}\right)$

Comments

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?

Answer 1

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.

Comments

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.

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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

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

Answer 2

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.

Answer 3

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.

Comments

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.

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Here i think,

P(W|X) = P(X|W)*P(W) /  P(X|W)*P(W) + P(X|W~)*P(W~)

and Match to win or loose is equally probable = ½.

Answer 4

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

 

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

 

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