Societal biases that are contained in retrieved documents have received increased interest. Such biases, which are often prevalent in the training data and learned by the model, can cause societal harms, by misrepresenting certain groups, and by enforcing stereotypes. Mitigating such biases demands algorithms that balance the trade-off between maximized utility for the user with fairness objectives, which incentivize unbiased rankings. Prior work on bias mitigation often assumes that ranking scores, which correspond to the utility that a document holds for a user, can be accurately determined. In reality, there is always a degree of uncertainty in the estimate of expected document utility. This uncertainty can be approximated by viewing ranking models through a Bayesian perspective, where the standard deterministic score becomes a distribution. In this work, we investigate whether uncertainty estimates can be used to decrease the amount of bias in the ranked results, while minimizing loss in measured utility. We introduce a simple method that uses the uncertainty of the ranking scores for an uncertainty-aware, post hoc approach to bias mitigation. We compare our proposed method with existing baselines for bias mitigation with respect to the utility-fairness trade-off, the controllability of methods, and computational costs. We show that an uncertainty-based approach can provide an intuitive and flexible trade-off that outperforms all baselines without additional training requirements, allowing for the post hoc use of this approach on top of arbitrary retrieval models.