Expected Value vs Expected Count

The expected value is the mean you would see over many repetitions of the random process, and it need not be a value the variable can actually take. For example, a game that pays 2 dollars with probability 0.5 and nothing with probability 0.5 has expected value $2(0.5) + 0(0.5) = 1$ dollar. In general it is written $E(X) = \mu_X = \sum x_i \, p_i$ (E of X, the mean of X, sums each value times its probability). It is the balance point of the probability distribution.

Expected counts are the tallies a chi-square model predicts for each cell when the null hypothesis holds, and the test measures how far the observed counts drift from them. For a two-way table, each cell's expected count is $E = \frac{(\text{row total}) \times (\text{column total})}{\text{grand total}}$. For example, in a 200-person table a cell whose row totals 80 and column totals 50 has $E = \frac{80 \times 50}{200} = 20$. Every expected count should be at least 5 for the chi-square approximation to be trustworthy.