  In this paper, we consider the problem of noiseless non-adaptive
probabilistic group testing, in which the goal is high-probability recovery of
the defective set. We show that in the case of $n$ items among which $k$ are
defective, the smallest possible number of tests equals $\min\{ C_{k,n} k \log
n, n\}$ up to lower-order asymptotic terms, where $C_{k,n}$ is a uniformly
bounded constant (varying depending on the scaling of $k$ with respect to $n$)
with a simple explicit expression. The algorithmic upper bound follows from a
minor adaptation of an existing analysis of the Definite Defectives (DD)
algorithm, and the algorithm-independent lower bound builds on existing works
for the regimes $k \le n^{1-\Omega(1)}$ and $k = \Theta(n)$. In sufficiently
sparse regimes (including $k = o\big( \frac{n}{\log n} \big)$), our main result
generalizes that of Coja-Oghlan {\em et al.} (2020) by avoiding the assumption
$k \le n^{1-\Omega(1)}$, whereas in sufficiently dense regimes (including $k =
\omega\big( \frac{n}{\log n} \big)$), our main result shows that individual
testing is asymptotically optimal for any non-zero target success probability,
thus strengthening an existing result of Aldridge (2019) in terms of both the
error probability and the assumed scaling of $k$.
