Sure you can generalize it by weighting the outcomes by their probabilities (not mentioned in the article). But how are you going to calculate those probabilities any more easily than you can solve the posed problem?

The more "intuitive" way of generalizing it (integer counting) does not work in general. I think the example is likely to be misleading to anyone who doesn't already understand this stuff.

As suggested by the footnote, the reason the given example can be solved elegantly is because of the symmetry.