A fair six-sided die is rolled. What is the expected value of the number rolled?

The expected value is a measure of the "average" or "central" value of an uncertain variable, in this case, the roll of a fair six-sided die. To find the expected value, we multiply each outcome of the random variable by their respective probabilities and sum these products.

A fair die has equal probabilities for each side, which means the probability of each side is $\frac{1}{6}$, since there are six sides.

The expected value of the number rolled on a fair six-sided die can be calculated using the following formula:

$$E(X) = \sum_{i=1}^{6}x_i P(x_i)\\ $$

Here, x_i are the possible outcomes (1, 2, 3, 4, 5, 6) and P(x_i) is the probability of each outcome, which is $\frac{1}{6}$.

Now, we plug the values into the formula:

$$E(X) = 1\left(\frac{1}{6}\right) + 2\left(\frac{1}{6}\right) + 3\left(\frac{1}{6}\right) + 4\left(\frac{1}{6}\right) + 5\left(\frac{1}{6}\right) + 6\left(\frac{1}{6}\right)\\ $$

$$E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)\\ $$

$$E(X) = \frac{1}{6}(21) \\ $$

E(X) = 3.5

So, the expected value of the number rolled on a fair six-sided die is 3.5. This means that over a large number of rolls, the average result is expected to be approximately 3.5.