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We introduce the concept of \emph{expected exposure} as the average attention ranked items receive from users over repeated samples of the same query. Furthermore, we advocate for the adoption of the principle of equal expected exposure: given a fixed information need, no item should receive more or less expected exposure than any other item of the same relevance grade. We argue that this principle is desirable for many retrieval objectives and scenarios, including topical diversity and fair ranking. Leveraging user models from existing retrieval metrics, we propose a general evaluation methodology based on expected exposure and draw connections to related metrics in information retrieval evaluation. Importantly, this methodology relaxes classic information retrieval assumptions, allowing a system, in response to a query, to produce a \emph{distribution over rankings} instead of a single fixed ranking. We study the behavior of the expected exposure metric and stochastic rankers across a variety of information access conditions, including \emph{ad hoc} retrieval and recommendation. We believe that measuring and optimizing expected exposure metrics using randomization opens a new area for retrieval algorithm development and progress.