Asymptotically almost all $\lambda$-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random $\lambda$-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the $\lambda$-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator