The relevant quote from the Wiki:

The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an actual setting things get a bit sticky. Just how do we know that "at least" one is a boy? One description of the problem states that we look into a window, see only one child and it is a boy. This sounds like the same assumption. However, this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn, ascertaining that it is a boy, then replacing). Let's call the statement "the sample is a boy" proposition "b". Now we have: P(BB|b) = P(b|BB) * P(BB) / P(b) = 1 * 1/4 / 1/2 = 1/2. The difference here is the P(b), which is just the probability of drawing a boy from all possible cases (i.e. without the "at least"), which is clearly 0.5. The Bayesian analysis generalizes easily to the case in which we relax the 50/50 population assumption. If we have no information about the populations then we assume a "flat prior", i.e. P(GG) = P(BB) = P(G.B) = 1/3. In this case the "at least" assumption produces the result P(BB|B) = 1/2, and the sampling assumption produces P(BB|b) = 2/3, a result also derivable from the Rule of Succession.

We have no general population information here. We have one man with at least one boy.