In human-agent interaction, one key challenge is the evaluation of the user's experience. In the virtual reality domain, the sense of presence and co-presence, reflecting the psychological immersion of the user, is generally assessed through well-grounded subjective post-experience questionnaires. In this article, we aim at presenting a new way to automatically predict the sense of presence and co-presence of a user at the end of an interaction based on specific verbal and non-verbal behavioral cues automatically computed. A random forest algorithm has been applied on a human-agent interaction corpus collected in the specific context of a virtual environment developed to train doctors to break bad news to a virtual patient. The performance of the models demonstrate the capacity to automatically and accurately predict the level of presence and co-presence, but also show the relevancy of the verbal and non-verbal behavioral cues as objective measures of presence.