We present two online experiments investigating trust in artificial intelligence (AI) as a primary and secondary medical diagnosis tool and one experiment testing two methods to increase trust in AI. Participants in Experiment 1 read hypothetical scenarios of low and high-risk diseases, followed by two sequential diagnoses, and estimated their trust in the medical findings. In three between-participants groups, the first and second diagnoses were given by: human and AI, AI and human, and human and human doctors, respectively. In Experiment 2 we examined if people expected higher standards of performance from AI than human doctors, in order to trust AI treatment recommendations. In Experiment 3 we investigated the possibility to increase trust in AI diagnoses by: (i) informing our participants that the AI outperforms the human doctor, and (ii) nudging them to prefer AI diagnoses in a choice between AI and human doctors. Results indicate overall lower trust in AI, as well as for diagnoses of high-risk diseases. Participants trusted AI doctors less than humans for first diagnoses, and they were also less likely to trust a second opinion from an AI doctor for high risk diseases. Surprisingly, results highlight that people have comparable standards of performance for AI and human doctors and that trust in AI does not increase when people are told the AI outperforms the human doctor. Importantly, we find that the gap in trust between AI and human diagnoses is eliminated when people are nudged to select AI in a free-choice paradigm between human and AI diagnoses, with trust for AI diagnoses significantly increased when participants could choose their doctor. These findings isolate control over one's medical practitioner as a valid candidate for future trust-related medical diagnosis and highlight a solid potential path to smooth acceptance of AI diagnoses amongst patients.