Steven Strogatz has a column up on why it’s easier to think about natural frequencies rather than conditional probabilities.
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An Ergodic Walk 
Great column!
I am a bit ticked by this column. I think that using the formula for conditional probability or computing frequencies as explained in the column are exactly the same thing. The difference is that the former applies a memorized formula without understanding it whereas the latter basically “derives” that formula every time a computation is needed. To claim that those are intrinsically different methods, or to discuss whether one is more valid than the other, shows a lack of understanding of what mathematics is.
I don’t think it’s a question of *mathematical* validity — you get the same number at the end. As a student, it may be more valuable to rederive the formula each time until you develop the intuition as to why it is true.
I remember learning how to square numbers by playing with grids of squares (10×10, 1×10, etc) and developing a geometric feeling for why (a + b)^2 = a^2 + 2ab + b^2. It’s not a more or less valid way for learning that formula, but the abstraction was easier to understand moving from physical manipulations to mental manipulations rather than the reverse.
I entirely agree that as a student is is more valuable to rederive the formula each time (which is what the “method of natural frequencies” is). I was taking issue with the article’s implicit statement that the two “methods” are not the same. To me, that means that they do not understand the formula for conditional probability.
This psychology article discusses reasons why using the formula and computing frequencies aren’t exactly the same thing. Here’s one passage from it:
It turns out that human performance in probabilistic reasoning tasks is remarkably sensitive to the format in which information is presented and answers asked for. Most experiments that elicited “non-normative” performance asked subjects to judge the probability of a single event (e.g., “What is the chance that a person who tests positive for the disease actually has it?”). However, many purported biases and fallacies disappear when people are asked to judge a frequency instead (e.g., “How many people who test positive for the disease will actually have it?”).
One of them may be easier to understand for a person, or “make more sense”, or what-have-you, but mathematically they are exactly the same thing. After all, probability means frequency.