The activation of a T cell is a stochastic process, and depends on the integrated strength of signals 1 and 2 resulting from its encounter with an antigen presenting cell. The net outcome of thymic selection and peripheral circulation over many such encounters in the presence of mechanisms for both central and peripheral tolerance is difficult to deduce by intuition alone. We therefore introduce a simple mathematical model that allows us to explore the roles and interaction of different thresholds, costimulation and anergy, as well as make predictions about expected immune system behaviour. We show that stochastic activation in the context of repeated encounters results in lowering the apparent activation threshold for T cells. This effect may contribute significantly to the efficiency of negative selection, although a low avidity subset of auto-reactive thymocytes can still be exported. A simple peripheral mechanism for peripheral tolerance is shown to be highly effective at dealing with this low avidity subset. Finally, the trade-offs between sensitivity and specificity are examined. © 2005 Elsevier B.V. All rights reserved.