Expected Value vs Probability Distribution

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.

A probability distribution shows how the total probability of $1$ is shared among a random variable's possible values. For example, rolling a fair six-sided die gives each value from 1 to 6 a probability of $\frac{1}{6}$. For a discrete variable the probabilities must satisfy $0 \le p_i \le 1$ and $\sum p_i = 1$ (each probability is between 0 and 1, and they add to 1). Continuous variables use a density curve instead, where probability is the area under the curve.