Two dice are thrown together , what is the probabability that the sum of the numbers on the two faces is neither divisible by 3 nor by 5.

Answer

To solve the problem of finding the probability that the sum of the numbers on two thrown dice is neither divisible by 3 nor by 5, we can follow these steps:

### Step 1: Determine the Sample Space
When two dice are thrown, each die has 6 faces. Therefore, the total number of outcomes when throwing two dice is:
\[ N_s = 6 \times 6 = 36 \]

### Step 2: Identify the Sums of Two Dice
The possible sums when rolling two dice range from 2 (1+1) to 12 (6+6). We need to find which of these sums are divisible by 3 or 5.

### Step 3: Identify Sums Divisible by 3
The sums that are divisible by 3 from the possible sums (2 to 12) are:
- 3 (1+2, 2+1)
- 6 (1+5, 2+4, 3+3, 4+2, 5+1)
- 9 (3+6, 4+5, 5+4, 6+3)
- 12 (6+6)

So, the sums divisible by 3 are: 3, 6, 9, and 12.

### Step 4: Identify Sums Divisible by 5
The sums that are divisible by 5 from the possible sums are:
- 5 (1+4, 2+3, 3+2, 4+1)
- 10 (4+6, 5+5, 6+4)

So, the sums divisible by 5 are: 5 and 10.

### Step 5: Combine the Results
Now, we need to combine the results to find the sums that are either divisible by 3 or 5:
- Divisible by 3: 3, 6, 9, 12
- Divisible by 5: 5, 10

The combined sums that are either divisible by 3 or 5 are: 3, 5, 6, 9, 10, and 12.

### Step 6: Count the Favorable Outcomes
Next, we need to count how many outcomes correspond to these sums:
- **Sum = 3:** (1,2), (2,1) → 2 outcomes
- **Sum = 5:** (1,4), (2,3), (3,2), (4,1) → 4 outcomes
- **Sum = 6:** (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
- **Sum = 9:** (3,6), (4,5), (5,4), (6,3) → 4 outcomes
- **Sum = 10:** (4,6), (5,5), (6,4) → 3 outcomes
- **Sum = 12:** (6,6) → 1 outcome

Adding these outcomes gives:
\[ 2 + 4 + 5 + 4 + 3 + 1 = 19 \text{ outcomes} \]

### Step 7: Calculate Outcomes Not Divisible by 3 or 5
To find the outcomes that are neither divisible by 3 nor by 5, we subtract the number of outcomes that are divisible by 3 or 5 from the total outcomes:
\[ \text{Outcomes not divisible by 3 or 5} = 36 - 19 = 17 \]

### Step 8: Calculate the Probability
Finally, the probability that the sum of the numbers on the two faces is neither divisible by 3 nor by 5 is given by:
\[ P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{17}{36} \]

### Final Answer
The probability that the sum of the numbers on the two faces is neither divisible by 3 nor by 5 is:
\[ \frac{17}{36} \]