  We consider the cardinality of supercritical oriented bond percolation in two
dimensions. We show that, whenever the origin is conditioned to percolate, the
process appropriately normalized converges asymptotically in distribution to
the standard normal law. This resolves a longstanding open problem pointed out
to in several instances in the literature. The result applies also to the
continuous-time analog of the process, viz. the basic one-dimensional contact
process. We also derive general random-indices central limit theorems for
associated random variables as byproducts of our proof.
