  Identification of defective members of large populations has been widely
studied in the statistics community under the name of group testing. It
involves grouping subsets of items into different pools and detecting defective
members based on the set of test results obtained for each pool.
  In a classical noiseless group testing setup, it is assumed that the sampling
procedure is fully known to the reconstruction algorithm, in the sense that the
existence of a defective member in a pool results in the test outcome of that
pool to be positive. However, this may not be always a valid assumption in some
cases of interest. In particular, we consider the case where the defective
items in a pool can become independently inactive with a certain probability.
Hence, one may obtain a negative test result in a pool despite containing some
defective items. As a result, any sampling and reconstruction method should be
able to cope with two different types of uncertainty, i.e., the unknown set of
defective items and the partially unknown, probabilistic testing procedure.
  In this work, motivated by the application of detecting infected people in
viral epidemics, we design non-adaptive sampling procedures that allow
successful identification of the defective items through a set of probabilistic
tests. Our design requires only a small number of tests to single out the
defective items. In particular, for a population of size $N$ and at most $K$
defective items with activation probability $p$, our results show that $M =
O(K^2\log{(N/K)}/p^3)$ tests is sufficient if the sampling procedure should
work for all possible sets of defective items, while $M = O(K\log{(N)}/p^3)$
tests is enough to be successful for any single set of defective items.
Moreover, we show that the defective members can be recovered using a simple
reconstruction algorithm with complexity of $O(MN)$.
