  We consider nonadaptive group testing with Bernoulli tests, where each item
is placed in each test independently with some fixed probability. We give a
tight threshold on the maximum number of tests required to find the defective
set under optimal Bernoulli testing. Achievability is given by a result of
Scarlett and Cevher; here we give a converse bound showing that this result is
best possible. Our new converse requires three parts: a typicality bound
generalising the trivial counting bound, a converse on the COMP algorithm of
Chan et al, and a bound on the SSS algorithm similar to that given by Aldridge,
Baldassini, and Johnson. Our result has a number of important corollaries, in
particular that, in denser cases, Bernoulli nonadaptive group testing is
strictly worse than the best adaptive strategies.
