  We study the probability that a random polynomial with integer coefficients
is reducible when factored over the rational numbers. Using computer-generated
data, we investigate a number of different models, including both monic and
non-monic polynomials. Our data supports conjectures made by Odlyzko and Poonen
and by Konyagin, and we formulate a universality heuristic and new conjectures
that connect their work with Hilbert's Irreducibility Theorem and work of van
der Waerden. The data indicates that the probability that a random polynomial
is reducible divided by the probability that there is a linear factor appears
to approach a constant and, in the large-degree limit, this constant appears to
approach one. In cases where the model makes it impossible for the random
polynomial to have a linear factor, the probability of reducibility appears to
be close to the probability of having a non-linear, low-degree factor. We also
study characteristic polynomials of random matrices with +1 and -1 entries.
