Safe hypothesis tests are tests that are robust under accumulation bias, namely when there are dependencies between the results of previous studies and the decision whether to conduct further studies. We construct two types of safe test for the 2 × 2 contingency table, the conditional and unconditional safe tests. In general safe tests are given by an information projection that may be difficult to compute. The conditional tests we construct however are given either in explicit form or implicitly via a defining equation. The same can be said of many of the unconditional tests we construct, for which we prove a number of theoretical results enabling their quick calculation when not given explicitly. The method we develop to accomplish this may perhaps be used to identify optimal safe tests in many other scenarios.