A lottery is a choice problem, where each outcome has a certain probability:

\begin{equation} [S_1:p_1, \dots, S_{n}:p_{n}] \end{equation}

where, \(S_{j}\) has \(p_{j}\) chance of occurring.

utility of a lottery

For a lottery, the utility thereof is the probability of a state happening times the utility of the state:

that is,

\begin{equation} U([S_1:p_1, \dots, S_{n}:p_{n}]) = \sum_{i=1}^{N} p_{i}U(S_{i})} \end{equation}

We have:

\begin{equation} || x ||_{p} = \qty(| x_1 |^{p} + \dots + | x_{n} |^{p})^{\frac{1}{p}} \end{equation}