**Expected Value of the Sum of Numbers on Two Unbiased Dice**

To find the expected value of the sum of numbers on two unbiased dice, we need to consider all the possible outcomes and their respective probabilities.

**Possible Outcomes:**

When two unbiased dice are thrown, each die can show a number from 1 to 6. Therefore, the possible outcomes for each die are:

1, 2, 3, 4, 5, 6

Since there are two dice, the total number of possible outcomes is the product of the number of outcomes for each die, which is 6 * 6 = 36.

**Calculating Probabilities:**

To calculate the probability of each outcome, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.

For example, when the two dice show a sum of 2, there is only one favorable outcome: (1, 1). Therefore, the probability of getting a sum of 2 is 1/36.

Similarly, when the sum is 3, the favorable outcomes are (1, 2) and (2, 1), so the probability is 2/36.

By calculating the probabilities for all possible outcomes, we obtain the following table:

Sum of Numbers | Number of Favorable Outcomes | Probability
-------------- | --------------------------- | -----------
2 | 1 | 1/36
3 | 2 | 2/36
4 | 3 | 3/36
5 | 4 | 4/36
6 | 5 | 5/36
7 | 6 | 6/36
8 | 5 | 5/36
9 | 4 | 4/36
10 | 3 | 3/36
11 | 2 | 2/36
12 | 1 | 1/36

**Calculating the Expected Value:**

The expected value of a random variable is the sum of each possible value multiplied by its respective probability.

To find the expected value of the sum of numbers on two unbiased dice, we calculate:

(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)

Simplifying this expression, we get:

2/36 + 6/36 + 12/36 + 20/36 + 30/36 + 42/36 + 40/36 + 36/36 + 30/36 + 22/36 + 12/36

Adding all the terms, we obtain the expected value of:

252/36 = 7

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