Sequences of random binary numbers created from polarization measurements of single photons were subjected to a comprehensive runs analysis. Photon pairs from a spontaneous parametric downconversion source were detected in coincidence, with one photon acting as a trigger while the other was analyzed for horizontal or vertical polarization. The resulting sequences of polarization measurements were tested for runs of consecutive vertical or horizontal outcomes against a theory of nonoverlapping runs, without numerical unbiasing. The sequences produced no statistically significant discrepancies with the predicted numbers of runs, even with multiphoton events retained.