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Random testing is increasingly popular and successful, but tends to spend most time rediscovering the ``most probable bugs'' again and again, reducing the value of long test runs on buggy software. We present a new automated method to adapt random test case generation so that already-discovered bugs are avoided, and further test effort can be devoted to searching for {\em new} bugs instead. We evaluate our method primarily against RANDOOP-style testing, in three different settings---our method avoids rediscovering bugs more successfully than RANDOOP and in some cases finds bugs that RANDOOP did not find at all.