Two unbiased dice are thrown . Find the probability that
sum of the numbers on the two faces is neither 9 nor 11 ,

Answer

To solve the problem of finding the probability that the sum of the numbers on two thrown unbiased dice is neither 9 nor 11, we can follow these steps:

### Step 1: Determine the total number of outcomes
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:

\[ \text{Total outcomes} = 6 \times 6 = 36 \]

### Step 2: Calculate the probability of getting a sum of 9
Next, we need to find the combinations that yield a sum of 9. The possible pairs (outcomes) that give a sum of 9 are:

- (3, 6)
- (4, 5)
- (5, 4)
- (6, 3)

Thus, there are 4 outcomes that result in a sum of 9.

The probability of getting a sum of 9 is calculated as:

\[ P(\text{sum of 9}) = \frac{\text{Number of favorable outcomes for sum of 9}}{\text{Total outcomes}} = \frac{4}{36} = \frac{1}{9} \]

### Step 3: Calculate the probability of getting a sum of 11
Now, we find the combinations that yield a sum of 11. The possible pairs that give a sum of 11 are:

- (5, 6)
- (6, 5)

Thus, there are 2 outcomes that result in a sum of 11.

The probability of getting a sum of 11 is calculated as:

\[ P(\text{sum of 11}) = \frac{\text{Number of favorable outcomes for sum of 11}}{\text{Total outcomes}} = \frac{2}{36} = \frac{1}{18} \]

### Step 4: Calculate the probability of getting a sum of either 9 or 11
To find the probability of getting a sum of either 9 or 11, we add the probabilities we calculated in the previous steps:

\[ P(\text{sum of 9 or 11}) = P(\text{sum of 9}) + P(\text{sum of 11}) = \frac{1}{9} + \frac{1}{18} \]

To add these fractions, we need a common denominator. The least common multiple of 9 and 18 is 18:

\[ P(\text{sum of 9}) = \frac{2}{18}, \quad P(\text{sum of 11}) = \frac{1}{18} \]
\[ P(\text{sum of 9 or 11}) = \frac{2}{18} + \frac{1}{18} = \frac{3}{18} = \frac{1}{6} \]

### Step 5: Calculate the probability that the sum is neither 9 nor 11
Finally, we find the probability that the sum is neither 9 nor 11 by subtracting the probability of getting a sum of 9 or 11 from 1:

\[ P(\text{neither 9 nor 11}) = 1 - P(\text{sum of 9 or 11}) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Final Answer
Thus, the probability that the sum of the numbers on the two faces is neither 9 nor 11 is:

\[ \frac{5}{6} \]