This paper considers the reasons mathematicians give for making probabilistic judgments about unproved mathematical statements, and discusses how one might interpret and justify such judgments more formally. Following Pólya, I argue that we update our probabilistic judgments in a broadly Bayesian way, while to explain what they mean in the first place, I argue that they are referring not so much to the truth of the statements as to the likely existence or otherwise of reasons for them. The link between (...) the two is provided by a "no-miracle" principle, which says that a surprising mathematical statement will not be true unless it is true for a reason. This principle applies only to statements that are sufficiently natural, so the paper also sets out criteria for a statement to be more or less natural. (shrink)