Introduction:
The sum of two dice is a classic example of a discrete probability distribution. The probability of each possible sum can be calculated by counting the number of ways to get that sum and dividing by the total number of possible outcomes.

Finding the Probability Distribution:
To find the probability distribution of the sum of two dice, we need to consider all the possible outcomes when rolling two dice. There are 36 possible outcomes in total, ranging from a sum of 2 to a sum of 12. We can count how many ways there are to get each sum and divide by 36 to get the probability.

Calculating the Mean:
The mean of a probability distribution is also known as its expected value. To calculate the mean of the sum of two dice, we need to multiply each possible sum by its probability and add up the results. This gives us the expected value of the sum.

Example Calculation:
Suppose we roll two dice. What is the probability distribution of the sum, and what is the expected value?

There are six possible outcomes for each die, so there are 6 x 6 = 36 possible outcomes when rolling two dice.

To get a sum of 2, we need to roll a 1 on both dice. There is only one way to do this, so the probability is 1/36.
To get a sum of 3, we can roll a 1 and a 2, or a 2 and a 1. There are two ways to do this, so the probability is 2/36 = 1/18.
Continuing in this way, we can calculate the probabilities for all the possible sums:

Sum Probability
2 1/36
3 2/36
4 3/36
5 4/36
6 5/36
7 6/36
8 5/36
9 4/36
10 3/36
11 2/36
12 1/36

To calculate the expected value, we need to multiply each sum by its probability and add up the results:

E(X) = (2 x 1/36) + (3 x 1/18) + (4 x 1/12) + (5 x 1/9) + (6 x 5/36) + (7 x 1/6) + (8 x 5/36) + (9 x 4/36) + (10 x 3/36) + (11 x 2/36) + (12 x 1/36)

Simplifying this expression, we get:

E(X) = 7

Therefore, the expected value of the sum of two dice is 7.

Conclusion:
In conclusion, the mean or expected value of the sum of two dice can be found by multiplying each possible sum by its probability and adding up the results. In the example above, the expected value was found to be 7.