To find the probability that the sum of the top faces of two dice, each having faces labeled `1, 2, 3, 5, 7, 11`, is less than or equal to `8`, we can follow these steps:
### Step 1: Determine the total number of outcomes
Each die has 6 faces, so when rolling two dice, the total number of outcomes is:
\[
\text{Total Outcomes} = 6 \times 6 = 36
\]
### Step 2: Identify the favorable outcomes
We need to find all the combinations of the two dice where the sum of the top faces is less than or equal to `8`. We will systematically check each possible outcome:
- **When the first die shows 1:**
- \(1 + 1 = 2\)
- \(1 + 2 = 3\)
- \(1 + 3 = 4\)
- \(1 + 5 = 6\)
- \(1 + 7 = 8\)
- \(1 + 11 = 12\) (not valid)
**Favorable outcomes:** (1,1), (1,2), (1,3), (1,5), (1,7) → 5 outcomes
- **When the first die shows 2:**
- \(2 + 1 = 3\)
- \(2 + 2 = 4\)
- \(2 + 3 = 5\)
- \(2 + 5 = 7\)
- \(2 + 7 = 9\) (not valid)
- \(2 + 11 = 13\) (not valid)
**Favorable outcomes:** (2,1), (2,2), (2,3), (2,5) → 4 outcomes
- **When the first die shows 3:**
- \(3 + 1 = 4\)
- \(3 + 2 = 5\)
- \(3 + 3 = 6\)
- \(3 + 5 = 8\)
- \(3 + 7 = 10\) (not valid)
- \(3 + 11 = 14\) (not valid)
**Favorable outcomes:** (3,1), (3,2), (3,3), (3,5) → 4 outcomes
- **When the first die shows 5:**
- \(5 + 1 = 6\)
- \(5 + 2 = 7\)
- \(5 + 3 = 8\)
- \(5 + 5 = 10\) (not valid)
- \(5 + 7 = 12\) (not valid)
- \(5 + 11 = 16\) (not valid)
**Favorable outcomes:** (5,1), (5,2), (5,3) → 3 outcomes
- **When the first die shows 7:**
- \(7 + 1 = 8\)
- \(7 + 2 = 9\) (not valid)
- \(7 + 3 = 10\) (not valid)
- \(7 + 5 = 12\) (not valid)
- \(7 + 7 = 14\) (not valid)
- \(7 + 11 = 18\) (not valid)
**Favorable outcomes:** (7,1) → 1 outcome
- **When the first die shows 11:**
- All combinations exceed 8.
**Favorable outcomes:** None → 0 outcomes
### Step 3: Count the total number of favorable outcomes
Now, we add up all the favorable outcomes:
\[
5 + 4 + 4 + 3 + 1 + 0 = 17
\]
### Step 4: Calculate the probability
The probability \(P\) that the sum of the top faces is less than or equal to `8` is given by:
\[
P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{17}{36}
\]
### Final Answer
The probability that the sum of the top faces is less than or equal to `8` is:
\[
\frac{17}{36}
\]
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