Randomised controlled trials (RCTs) play a central role in modern medical advance, and they require participants who understand and accept the procedures involved. Published evidence suggests that RCT participants often fail to understand that treatments are allocated at random and that clinicians are in equipoise about which treatment is best. We examine background assumptions that members of the public might draw upon if invited to take part in a RCT. Four studies (N=82; 67; 67; 128), in the UK, identified whether members of the public (i). accept that an individual clinician might be genuinely unsure which of two treatments was better; (ii). judge that when there is uncertainty it is acceptable to suggest deciding at random; (iii). recognise scientific benefits of random allocation to treatment conditions in a trial. Around half the participants were loathe to accept that a clinician could be completely uncertain, and this was no different whether the context was one of individual treatment or research. Most participants found it unacceptable to suggest allocating treatment at random, though there was weak evidence that a research context may reduce the unacceptability. Participants did not judge that more certain knowledge would be gained about which treatment was best when treatments were allocated at random rather than by patient/doctor choice: scientific benefits of randomisation were apparently not recognised. Judgements were no different in non-medical contexts. Results suggest a large mismatch between the assumptions underlying the trial design, and the assumptions that lay participants can bring to bear when they try to make sense of descriptive information about randomisation and equipoise. Previous attempts to improve understanding by improving the clarity or salience of trial information, or of making explicit the research context, while helpful, may need to be supplemented with accessible explanations for random allocation.