# Classical Probability Approach

The definition of classical probability is

Classical Probability: If a random experiment can result in $n$ mutually exclusive and equally likely outcomes and if $n_A$ of these outcomes have an attribute $A$, then the probability of $A$ is the fraction $n_A/n.$

In the book Mood, A. M., Graybill, F. A., & Boes, D. C. (1974). Introduction to the Theory of Statistics 1974. McGraw-Hill Kogakusha, it is written that

difficulty with the classical approach is encounterd when we try to answer the question: What is the probability that a child born in Chicago will be a boy?

Why can't classical definition answer it? Isn't it simply $0.5$, as the child's being a male or female is equally likely?

• The first quotation isn't a definition at all: it's a formula for a model. Should you choose to apply it, your first duty is to demonstrate that it is at least approximately correct--and that puts you right back to the beginning: what is "probability" in the first place and how do you measure it?
– whuber
Commented Jun 18, 2018 at 13:29

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.)

I think the classical definition is an idealization. In the real world things are very rarely so tidy. Child birth rates are actually a great example. At first glance, people tend to think they are 50-50, but they are not! More males are conceived (one theory is that sperm with the smaller Y chromosomes are slightly faster). More males are miscarried, but not enough to completely undo the 'advantage.'