This word you used, "independent" has a term in these kind of probabilistic systems, its called "memory". A system is memoryless if the next outcome is not dependent on previous outcomes. A coin flip is such a memoryless system. The odds it will rain on a particular day is not independent of pervious outcomes; such a system is not memoryless. All the examples in your comment share this property and so the reasoning doesn't strictly apply to memoryless systems.

The author is not making this point, and early on, he calls coin flips "Markovian" which is synonymous with "memoryless". Independence is assumed in this article.