Firstly, you should not confuse the philosophical issue of defining the meaning of probability, with the statistical issue of estimating the probability of a particular event. Merely defining the meaning of probability does not allow you to deduce the probability of a particular event. The only way to assess the probability that a child born in Chicago is a boy is to look at data on sex-ratios at birth. As you can see from this data, it is standard for there to be more males at birth than females, and hence it is unlikely to be true that the a birth of either sex is equally likely in Chicago. Hopefully this kind of case alerts us to the danger of making a priori statements about probabilities of events, without looking for empirical data.
In relation to the classical formulation of probability, you will notice that it is a count-based method that relies on a pre-existing pre-probabilistic conceptual notion of what is "equally likely". This is a major drawback of the classical formulation, since it defines probability in terms of a preliminary concept that is arguably a probabilistic determination. In the quote you supply, the difficulty with assessing the probability of this event is that there is not a clear class of "equally likely outcomes" that can be used as a basis for the count. The data we have show that male and female births are not equally likely. (And in any case, if we are going to try to determine the probability of a male birth, we should not beg the question by assuming the answer a priori.)