  In the bin covering problem, the goal is to fill as many bins as possible up
to a certain minimal level with a given set of items of different sizes. Online
variants, in which the items arrive one after another and have to be packed
immediately on their arrival without knowledge about the future items, have
been studied extensively in the literature. We study the simplest possible
online algorithm Dual Next-Fit, which packs all arriving items into the same
bin until it is filled and then proceeds with the next bin in the same manner.
The competitive ratio of this and any other reasonable online algorithm is
$1/2$.
  We study Dual Next-Fit in a probabilistic setting where the item sizes are
chosen i.i.d.\ according to a discrete distribution and we prove that, for
every distribution, its expected competitive ratio is at least $1/2+\epsilon$
for a constant $\epsilon>0$ independent of the distribution. We also prove an
upper bound of $2/3$ and better lower bounds for certain restricted classes of
distributions. Finally, we prove that the expected competitive ratio equals,
for a large class of distributions, the random-order ratio, which is the
expected competitive ratio when adversarially chosen items arrive in uniformly
random order.
