Two arbitrary lines in a 2D plane will meet with probability 1.0.

Two arbitrary lines in 3D space will meet with probability 0.0.

(In each case, the exceptions are vanishingly few relative to the norm.)

From a mathematical point of view, this isn't actually true (although it is intuitive). The probability that two arbitrary lines in 2D space will meet is undefined. Going back to basics, there are an unbounded number of possible lines. Without loss of generality, select y=0. (We don't lose generality because we can translate, rotate, and scale the plane to make any other selection equivalent to y=0.) Then the probability that the second line selected meets the first is defined as the number of lines which meet it divided by the universe of lines. Both sets are unbounded, so the probability is undefined (infinity / infinity).

If you'd like to resort to an argument based on transfinite numbers, I don't think it's valid. But it still undercuts the statement. It turns out that the two sets are of equivalent (transfinite) cardinality[1], so I suppose you could argue that the probability is still 1.0. But the same argument applies to arbitrary lines in 3D, 4D, or N-D space[2], so those are also 1.0.

Anybody see any problems there? I'm not a serious math geek, just a professor of computer science.

[1] The set of lines y=mx+b which don't meet y=0 are all pairs m=0, b!=0. The cardinality of that set is equal to the cardinality of the reals. The lines which do meet y=0 are everything else, and has cardinality equal to the set of all pairs of reals. It's fairly straightforward to show by, e.g., interleaving digits of our pair values, that there's an injective function from the pairs of reals to the reals. By inclusion, there's also an injective function from the singleton to the pairs. By the Cantor-Berstein-Schroeder Theorem, this means that a bijective function exists and the sets are of equivalent cardinality.

[2] Lines in N-D space are defined by a set of N real coefficients. We've already shown that the cardinality of the set of reals is equal to the cardinality of the set of pairs of reals. By induction on the number of coefficients, the cardinality of lines in N-D space is also the same as the cardinality of the reals.