  Despite the extreme diversity of T cell repertoires, many identical T-cell
receptor (TCR) sequences are found in a large number of individual mice and
humans. These widely-shared sequences, often referred to as `public', have been
suggested to be over-represented due to their potential immune functionality or
their ease of generation by V(D)J recombination. Here we show that even for
large cohorts the observed degree of sharing of TCR sequences between
individuals is well predicted by a model accounting for by the known
quantitative statistical biases in the generation process, together with a
simple model of thymic selection. Whether a sequence is shared by many
individuals is predicted to depend on the number of queried individuals and the
sampling depth, as well as on the sequence itself, in agreement with the data.
We introduce the degree of publicness conditional on the queried cohort size
and the size of the sampled repertoires. Based on these observations we propose
a public/private sequence classifier, `PUBLIC' (Public Universal Binary
Likelihood Inference Classifier), based on the generation probability, which
performs very well even for small cohort sizes.
