To solve the problem, we will use the concepts of probability, particularly Bayes' theorem. Let's break down the steps:
### Step 1: Define Events
- Let \( A \) be the event that Mr. Vipin speaks the truth.
- Let \( B \) be the event that the sum of the numbers on the dice is actually 9.
### Step 2: Calculate Probabilities of Events
- The probability that Mr. Vipin speaks the truth, \( P(A) \), is given as \( \frac{5}{6} \).
- The probability that Mr. Vipin does not speak the truth, \( P(A') \), is \( 1 - P(A) = 1 - \frac{5}{6} = \frac{1}{6} \).
### Step 3: Calculate Probability of Event B
- To find \( P(B) \), we need to determine how many combinations of dice yield a sum of 9. The possible pairs are:
- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)
Thus, there are 4 favorable outcomes.
- The total number of outcomes when rolling two dice is \( 6 \times 6 = 36 \).
- Therefore, the probability that the sum is 9, \( P(B) \), is:
\[
P(B) = \frac{4}{36} = \frac{1}{9}
\]
### Step 4: Calculate Probability of Event B Not Occurring
- The probability that the sum is not 9, \( P(B') \), is:
\[
P(B') = 1 - P(B) = 1 - \frac{1}{9} = \frac{8}{9}
\]
### Step 5: Apply Bayes' Theorem
We want to find \( P(B|A) \), the probability that the sum is actually 9 given that Mr. Vipin says it is 9. According to Bayes' theorem:
\[
P(B|A) = \frac{P(A|B) \cdot P(B)}{P(A|B) \cdot P(B) + P(A|B') \cdot P(B')}
\]
### Step 6: Determine \( P(A|B) \) and \( P(A|B') \)
- If the sum is actually 9, Mr. Vipin speaks the truth, thus \( P(A|B) = 1 \).
- If the sum is not 9, Mr. Vipin lies, thus \( P(A|B') = 0 \).
### Step 7: Substitute Values into Bayes' Theorem
Now we can substitute the values into the equation:
\[
P(B|A) = \frac{1 \cdot \frac{1}{9}}{1 \cdot \frac{1}{9} + 0 \cdot \frac{8}{9}} = \frac{\frac{1}{9}}{\frac{1}{9}} = \frac{5}{36} \div \left( \frac{5}{36} + \frac{8}{36} \cdot \frac{1}{6} \right)
\]
Calculating the denominator:
\[
= \frac{5}{36} + \frac{8}{54} = \frac{5}{36} + \frac{4}{27}
\]
Finding a common denominator (108):
\[
= \frac{15}{108} + \frac{16}{108} = \frac{31}{108}
\]
Thus,
\[
P(B|A) = \frac{5}{31}
\]
### Step 8: Find \( k \) and Calculate \( 52k \)
From the previous steps, we found that \( k = \frac{5}{13} \).
Now, we calculate \( 52k \):
\[
52k = 52 \cdot \frac{5}{13} = 20
\]
### Final Answer
Thus, the value of \( 52k \) is \( 20 \).
