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?a)1/2b)2/3c)3/4d)5/6e)6/7Correct answer is option 'E'. Can you explain this answer?

Question

Answer

Given Information:
- Karan tells the truth with a probability of 1/3 and lies with a probability of 2/3.
- Arjun tells the truth with a probability of 3/4 and lies with a probability of 1/4.

To Find:
The probability of India winning the cricket match given that Arjun says India won and Karan says India lost.

Approach:
1. Let's assume that the event 'India won' is denoted by A, and the event 'India lost' is denoted by A'.
2. We need to find the conditional probability of A given that Arjun says A and Karan says A'.

Solution:
We can find the probability using Bayes' theorem, which states:
P(A|B) = (P(B|A) * P(A)) / P(B)

Here:
- B represents the event that Arjun says India won and Karan says India lost.
- P(B|A) represents the probability that Arjun says India won and Karan says India lost, given that India actually won.
- P(A) represents the probability of India winning.
- P(B) represents the probability that Arjun says India won and Karan says India lost.

Let's calculate each of these probabilities step by step:

Step 1: Calculate P(B|A)
- We know that Arjun tells the truth with a probability of 3/4, so the probability that Arjun says India won given that India actually won is 3/4.
- Similarly, Karan lies with a probability of 2/3, so the probability that Karan says India lost given that India actually won is 2/3.
- Since Arjun and Karan's statements are independent, we can multiply their probabilities: P(B|A) = (3/4) * (2/3) = 1/2

Step 2: Calculate P(A)
- P(A) represents the probability of India winning, which we need to find.
- Let's assume the probability of India winning is x.
- Since there are only two possible outcomes (India winning or losing), the probability of India losing is 1 - x.
- We know that Karan tells the truth with a probability of 1/3, so the probability that Karan says India lost given that India actually lost is 1/3.
- Similarly, Arjun lies with a probability of 1/4, so the probability that Arjun says India won given that India actually lost is 1/4.
- Since Arjun and Karan's statements are independent, we can multiply their probabilities: P(A') = (1/4) * (1/3) = 1/12
- Since P(A) + P(A') = 1, we can write x + 1/12 = 1, and solve for x: x = 11/12

Step 3: Calculate P(B)
- P(B) represents the probability that Arjun says India won and Karan says India lost.
- Since Arjun and Karan's statements are independent, we can multiply their probabilities: P(B) = (3/4) * (2/3) = 1/2

Step 4: Calculate

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