The Tutte-Gr\"othendieck polynomial T(G;x,y) of a graph G
encodes numerous interesting combinatorial quantities associated
with the graph. Its evaluation in various points in the (x,y)
plane give the number of spanning forests of the graph, the number
of its strongly connected orientations, the number of its proper
k-colorings, the (all terminal) reliability probability of the
graph, and various other invariants the exact computation of each
of which is well known to be \#P-hard. Here we develop a general
technique that supplies fully polynomial randomised approximation
schemes for approximating the value of T(G;x,y) for any dense
graph G, that is, any graph on
n vertices whose minimum degree is \Omega(n), whenever
x \geq 1 and y > 1, and in various additional points.
Annan \cite{1} has dealt with the case y=1, x\geq 1.
This region includes evaluations of reliability and partition
functions of the ferromagnetic Q-state Potts model.
Extensions to linear matroids where T specialises to the
weight enumerator of linear codes are considered as well.