The specification of state machine replication (SMR) has no requirement on the final total order of commands. In blockchains based on SMR, however, order matters, since different orders could provide their clients with different financial rewards. Ordered consensus augments the specification of SMR to include specific guarantees on such order, with a focus on limiting the influence of Byzantine nodes. Real-world ordering manipulations, however, can and do happen even without Byzantine replicas, typically because of factors, such as faster networks or closer proximity to the blockchain infrastructure, that give some clients an unfair advantage. To address this challenge, this paper proceeds to extend ordered consensus by requiring it to also support equal opportunity, a concrete notion of fairness, widely adopted in social sciences. Informally, equal opportunity requires that two candidates who, according to a set of criteria deemed to be relevant, are equally qualified for a position (in our case, a specific slot in the SMR total order), should have an equal chance of landing it. We show how randomness can be leveraged to keep bias in check, and, to this end, introduce the secret random oracle (SRO), a system component that generates randomness in a fault-tolerant manner. We describe two SRO designs based, respectively, on trusted hardware and threshold verifiable random functions, and instantiate them in Bercow, a new ordered consensus protocol that, by approximating equal opportunity up to within a configurable factor, can effectively mitigate well-known ordering attacks in SMR-based blockchains.