The conventional test-per-scan built-in self-test (BIST) scheme needs a number of shift cycles followed by one capture cycle. Fault effects received by the scan flipflops are shifted out while shifting in the next test vector like scan testing. Unlike deterministic testing, it is unnecessary to apply a complete test vector to the scan chains. A new scan-based BIST scheme is proposed by properly controlling the test signals of the scan chains. Different biased random values are assigned to the test signals of scan flip-flops in separate scan chains. Capture cycles can be inserted at any clock cycle if necessary. A new testability estimation procedure according to the proposed testing scheme is presented. A greedy procedure is proposed to select a weight for each scan chain. Experimental results show that the proposed method can improve test effectiveness of scan-based BIST greatly, and most circuits can obtain complete fault coverage or very close to complete fault coverage. random testability, scan-based BIST, test signal, biased random testing