  Successful sequencing experiments require judicious sample selection.
However, this selection must often be performed on the basis of limited
preliminary data. Predicting the statistical properties of the final sample
based on preliminary data can be challenging, because numerous uncertain model
assumptions may be involved. Here, we ask whether we can predict ``omics"
variation across many samples by sequencing only a fraction of them. In the
infinite-genome limit, we find that a pilot study sequencing $5\%$ of a
population is sufficient to predict the number of genetic variants in the
entire population within $6\%$ of the correct value, using an estimator
agnostic to demography, selection, or population structure. To reach similar
accuracy in a finite genome with millions of polymorphisms, the pilot study
would require about $15\%$ of the population. We present computationally
efficient jackknife and linear programming methods that exhibit substantially
less bias than the state of the art when applied to simulated data and
sub-sampled 1000 Genomes Project data. Extrapolating based on the NHLBI Exome
Sequencing Project data, we predict that $7.2\%$ of sites in the capture region
would be variable in a sample of $50,000$ African-Americans, and $8.8\%$ in a
European sample of equal size. Finally, we show how the linear programming
method can also predict discovery rates of various genomic features, such as
the number of transcription factor binding sites across different cell types.
