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32 votes
32 votes

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. $\left(\dfrac{6}{7}\right)$
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6 Answers

Best answer
24 votes
24 votes
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.
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16 votes
16 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.

7 votes
7 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.

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

 

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

 

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

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