To solve this, let’s go through each part:

1. Constructing the Probability Distribution:

   • Since each die has 7 faces, numbered from 1 to 7, the possible sums when two 7-sided dice are rolled range from $2$ (1+1) to $14$ (7+7).
   • The number of outcomes for each sum can be calculated by counting the pairs that produce each sum.

2. Counting Outcomes for Each Sum:

   • There are $49$ total outcomes (since each die has 7 faces, and $7 \times 7 = 49$ combinations).
   • Let's count the occurrences for each sum.

Probability Distribution Table

Below is the distribution, where each entry shows the possible sum and the number of ways that sum can occur:

Sum	Occurrences	Probability (Occurrences/49)
2	1	$\frac{1}{49}$
3	2	$\frac{2}{49}$
4	3	$\frac{3}{49}$
5	4	$\frac{4}{49}$
6	5	$\frac{5}{49}$
7	6	$\frac{6}{49}$
8	7	$\frac{7}{49}$
9	6	$\frac{6}{49}$
10	5	$\frac{5}{49}$
11	4	$\frac{4}{49}$
12	3	$\frac{3}{49}$
13	2	$\frac{2}{49}$
14	1	$\frac{1}{49}$

Calculating the Mean

The mean $\mu$ of a probability distribution is calculated by: $\mu = \sum (x \cdot P(x))$ where $x$ is the sum and $P(x)$ is its probability.

Calculating each product: $\mu = 2 \cdot \frac{1}{49} + 3 \cdot \frac{2}{49} + 4 \cdot \frac{3}{49} + 5 \cdot \frac{4}{49} + 6 \cdot \frac{5}{49} + 7 \cdot \frac{6}{49} + 8 \cdot \frac{7}{49} + 9 \cdot \frac{6}{49} + 10 \cdot \frac{5}{49} + 11 \cdot \frac{4}{49} + 12 \cdot \frac{3}{49} + 13 \cdot \frac{2}{49} + 14 \cdot \frac{1}{49}$

Evaluating this gives: $\mu = \frac{2 + 6 + 12 + 20 + 30 + 42 + 56 + 54 + 50 + 44 + 36 + 26 + 14}{49} = \frac{392}{49} = 8$

Conclusion

• Mean (Expected Value): The mean of the sum when two 7-sided dice are rolled is $8$.

Would you like further details on the calculations?

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Here are some related questions to explore further:

1. What is the variance of this probability distribution?
2. How would the distribution change if one die had only 6 faces?
3. What is the probability of rolling an odd sum with these dice?
4. How does the mean compare if we used standard 6-sided dice instead?
5. Can we find the mode of this distribution?

Tip: Remember, when constructing a probability distribution for dice, consider each unique outcome carefully, as non-standard dice change typical patterns.