  Earlier work introduced a geometrically natural probability measure on the
group of all M\"obius transformations of the hyperbolic plane so as to be able
to study "random" groups of M\"obius transformations, and in particular random
two-generator groups. Here we extend these results to consider random punctured
tori. These Riemann surfaces have finite hyperbolic area $2\pi$ and fundamental
group the free group of rank 2. They can be obtained by pairing (identifying)
the opposite sides of an ideal hyperbolic quadrilateral. There is a natural
distribution on ideal quadrilateral given by the cross ratio of their vertices.
We identify this distribution and then calculate the distributions of various
geometric quantities associated with random punctured tori such as the base of
the geodesic length spectrum and the conformal modulus, along with more subtle
things such as the distribution of the distance in Teichm\"uller space to the
central "square" punctured torus.
