The glow worms are still distributed randomly in the mathematical sense, but knowing the position of one tells you something about the likely positions of others, so they are not independent.
Common parlance doesn't do a great job at talking about the features of random distributions, but when people say "purely random," they often seem to mean "uniformly distributed and independent." Both pictures have uniformly distributed points, but the glow worms are not independent.
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The concept you're looking for is "independence"
http://en.wikipedia.org/wiki/Independence_(probability_theor...
The glow worms are still distributed randomly in the mathematical sense, but knowing the position of one tells you something about the likely positions of others, so they are not independent.
Common parlance doesn't do a great job at talking about the features of random distributions, but when people say "purely random," they often seem to mean "uniformly distributed and independent." Both pictures have uniformly distributed points, but the glow worms are not independent.