Expected utility is the expected value in terms of the utility produced by an action. Each possible consequence of the action is assigned a utility by a utility function and weighed by the probability of that outcome occurring. The sum of these weighed utilities is defined to be the action's expected utility.
Von Neumann and Morgenstern proved the expected utility theorem, which says that when a rational agent chooses between different "gambles" (probability distributions over outcomes), the utility of such a gamble can always be seen as the expected utility of the gamble's outcome.
Humans, of course, are a different story.
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