The expected value of an action is the [-mean] numerical outcome of the possible results weighted by their Probability. It may actually be impossible to get the expected value, for example, if a coin toss decides between you getting \$0 and \$10, then we say you get "\$5 in expectation" even though there is no way for you to get \$5.
The expectation of V (often shortened to "the expected V") is how much V you expect to get on average. For example, the expectation of a payoff, or an expected payoff, is how much money you will get on average; the expectation of the duration of a speech, or an expected duration, is how long the speech will last "on average."
Suppose V has discrete possible values, say $~$V = x_{1},$~$ or $~$V = x_{2}, …, $~$ or $~$V = x_{k}$~$. Let $~$P(x_{i})$~$ refer to the probability that $~$V = x_{i}$~$. Then the expectation of V is given by:
$$~$\sum_{i=1}^{k}x_{i}P(x_{i})$~$$
Suppose V has continuous possible values, x. For instance, let $~$x \in \mathbb{R}$~$. Let $~$P(x)$~$ be the continuous probability distribution, or $~$\lim_{dx \to 0}$~$ of the probability that $~$x<V<(x+dx)$~$ divided by $~$dx$~$. Then the expectation of V is given by:
$$~$\int_{-∞}^{∞}xP(x)dx$~$$
Importance
A common principle of reasoning under uncertainty is that if you are trying to achieve a good G, you should choose the act that maximizes the expectation of G.