Definition:Consequence Function with Probability

Definition
Let $G$ be a game.

Let $P$ be a player of $G$.

Let $A$ be the set of moves available to $P$.

Let $C$ be the set of consequences of those moves.

Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves.

A consequence function for $P$ is a mapping from $A \times \Omega$ to $C$:
 * $g: A \times \Omega \to C$

interpreted as that $g \left({a, \omega}\right)$ is the consequence when the move is $a \in A$ and the realization $\omega$ of the random variable is $\omega \in \Omega$.