  The problem of Group Testing is to identify defective items out of a set of
objects by means of pool queries of the form "Does the pool contain at least a
defective?". The aim is of course to perform detection with the fewest possible
queries, a problem which has relevant practical applications in different
fields including molecular biology and computer science. Here we study GT in
the probabilistic setting focusing on the regime of small defective probability
and large number of objects, $p \to 0$ and $N \to \infty$. We construct and
analyze one-stage algorithms for which we establish the occurrence of a
non-detection/detection phase transition resulting in a sharp threshold, $\bar
M$, for the number of tests. By optimizing the pool design we construct
algorithms whose detection threshold follows the optimal scaling $\bar M\propto
Np|\log p|$. Then we consider two-stages algorithms and analyze their
performance for different choices of the first stage pools. In particular, via
a proper random choice of the pools, we construct algorithms which attain the
optimal value (previously determined in Ref. [16]) for the mean number of tests
required for complete detection. We finally discuss the optimal pool design in
the case of finite $p$.
