A fair die is rolled. What is the expected value of the number that shows up?

Step 1 ：We first need to know the probability distribution of the outcomes. For a fair die, the outcomes are equally likely. So for each outcome from 1 to 6, the probability is \(\frac{1}{6}\).

Step 2 ：The expected value is calculated by multiplying each outcome by its probability and then summing these products. So the expected value E is given by the formula: \[E = (1)\left(\frac{1}{6}\right)+(2)\left(\frac{1}{6}\right)+(3)\left(\frac{1}{6}\right)+(4)\left(\frac{1}{6}\right)+(5)\left(\frac{1}{6}\right)+(6)\left(\frac{1}{6}\right)\]

Step 3 ：We carry out the multiplication and addition: \[E = \frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}+\frac{6}{6} = \frac{21}{6}\]