Mr. Vipin, a famous liar, is known to speak the truth 5 out of 6 times. His blind folded friend Shubham throws a pair of dice and asked Vipin the result, who says the sum of numbers on the pair of disc is 9. The probability that the sum of numbers on the pair of dice is actually 9 is k, then the value of 52k is equal to

Answer

To solve the problem, we will use Bayes' theorem to find the probability that the sum of the numbers on the pair of dice is actually 9, given that Mr. Vipin claims it is 9.

**Step 1: Define the events**
- Let \( A \) be the event that Mr. Vipin tells the truth.
- Let \( B \) be the event that the sum of the numbers on the pair of dice is 9.

**Step 2: Find the probabilities of the events**
- The probability that Mr. Vipin tells the truth is given as:
\[ P(A) = \frac{5}{6} \]
- The probability that he lies is:
\[ P(A') = 1 - P(A) = \frac{1}{6} \]

**Step 3: Calculate the probability of event \( B \)**
- The total number of outcomes when throwing two dice is \( 6 \times 6 = 36 \).
- The combinations that yield a sum of 9 are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

This gives us 4 favorable outcomes. Therefore, the probability that the sum is 9 is:
\[ P(B) = \frac{4}{36} = \frac{1}{9} \]
- The probability that the sum is not 9 is:
\[ P(B') = 1 - P(B) = 1 - \frac{1}{9} = \frac{8}{9} \]

**Step 4: Calculate the conditional probabilities**
- If the sum is actually 9, the probability that Vipin tells the truth (i.e., he says the sum is 9) is:
\[ P(B|A) = 1 \]
- If the sum is not 9, the probability that Vipin lies (i.e., he says the sum is 9) is:
\[ P(B|A') = 0 \]

**Step 5: Apply Bayes' theorem**
We want to find \( P(B|A) \), which can be expressed using Bayes' theorem as:
\[ P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A|B) \cdot P(B) + P(A|B') \cdot P(B')} \]

Substituting the known values:
- \( P(A|B) = P(A) = \frac{5}{6} \)
- \( P(A|B') = P(A') = \frac{1}{6} \)

Thus, we have:
\[ P(B|A) = \frac{\left(\frac{5}{6}\right) \cdot \left(\frac{1}{9}\right)}{\left(\frac{5}{6} \cdot \frac{1}{9}\right) + \left(\frac{1}{6} \cdot \frac{8}{9}\right)} \]

Calculating the denominator:
\[ = \frac{5}{54} + \frac{8}{54} = \frac{13}{54} \]

So, we get:
\[ P(B|A) = \frac{\frac{5}{54}}{\frac{13}{54}} = \frac{5}{13} \]

**Step 6: Find the value of \( 52k \)**
Let \( k = P(B|A) = \frac{5}{13} \). Therefore:
\[ 52k = 52 \cdot \frac{5}{13} = 4 \cdot 5 = 20 \]

Thus, the final answer is:
\[ \boxed{20} \]