Evaluating query predicates on data samples is the only way to estimate their selectivity in certain scenarios. Finding a guaranteed optimal query plan is not a reasonable optimization goal in those cases as it might require an infinite number of samples. We therefore introduce probably approximately optimal query optimization (PAO) where the goal is to find a query plan whose cost is near-optimal with a certain probability. We will justify why PAO is a suitable formalism to model scenarios in which predicate sampling and optimization need to be interleaved. We present the first algorithm for PAO. Our algorithm is non-intrusive and uses standard query optimizers and sampling components as sub-functions. It is generic and can be applied to a wide range of scenarios. Our algorithm is iterative and calculates in each iteration a query plan together with a region in the selectivity space where the plan has near-optimal cost. It determines the confidence that the true selectivity values fall within the aforementioned region and chooses the next samples to take based on the current state if the confidence does not reach the threshold specified as problem input. We devise different algorithm variants and analyze their complexity. We experimentally compare them in terms of the number of optimizer invocations, samples, and iterations over many different query classes.