To solve the problem, we need to determine the probability that neither Tibu nor Babu wins the game when both roll a pair of dice. Tibu wins if the sum of the numbers on the dice is a prime number, and Babu wins if the product of the numbers on the dice is a prime number.
### Step-by-Step Solution:
1. **Understanding the Total Outcomes**:
- When rolling two dice, each die has 6 faces. Therefore, the total number of outcomes when rolling two dice is \(6 \times 6 = 36\).
2. **Finding the Probability of Tibu Winning (Sum is Prime)**:
- The possible sums when rolling two dice range from 2 to 12.
- The prime numbers in this range are: 2, 3, 5, 7, 11.
- Now, we calculate the combinations that yield these sums:
- Sum = 2: (1,1) → 1 way
- Sum = 3: (1,2), (2,1) → 2 ways
- Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 ways
- Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways
- Sum = 11: (5,6), (6,5) → 2 ways
- Total ways for Tibu to win: \(1 + 2 + 4 + 6 + 2 = 15\).
- Probability of Tibu winning \(P(A) = \frac{15}{36}\).
3. **Finding the Probability of Babu Winning (Product is Prime)**:
- The possible products when rolling two dice can be prime numbers: 2, 3, 5.
- Now, we calculate the combinations that yield these products:
- Product = 2: (1,2), (2,1) → 2 ways
- Product = 3: (1,3), (3,1) → 2 ways
- Product = 5: (1,5), (5,1) → 2 ways
- Total ways for Babu to win: \(2 + 2 + 2 = 6\).
- Probability of Babu winning \(P(B) = \frac{6}{36}\).
4. **Finding the Probability of Both Winning (Intersection)**:
- We need to find the cases where both Tibu and Babu win simultaneously:
- The only case where both the sum is prime and the product is prime is when the dice show (1,2) or (2,1).
- Total ways for both to win: \(2\).
- Probability of both winning \(P(A \cap B) = \frac{2}{36}\).
5. **Finding the Probability of Either Winning (Union)**:
- Using the formula for the probability of the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
- Substituting the values:
\[
P(A \cup B) = \frac{15}{36} + \frac{6}{36} - \frac{2}{36} = \frac{19}{36}
\]
6. **Finding the Probability that No One Wins**:
- The probability that neither Tibu nor Babu wins is given by:
\[
P(A^c \cap B^c) = 1 - P(A \cup B)
\]
- Substituting the value:
\[
P(A^c \cap B^c) = 1 - \frac{19}{36} = \frac{17}{36}
\]
### Final Answer:
The probability that no one wins is \(\frac{17}{36}\).
