This paper focuses on the problem of bounding fault detection probabilities in combinational circuits. Two algorithms, the complete cutting algorithm and the gate blocking algorithm, are presented that always produce true lower bounds on the detection probability of a fault. Both algorithms can be used to identify difficult-to-test faults and to quickly construct test sets for specific faults. Both algorithms have qualitative versions which provide insight into a circuit while avoiding arithmetic calculation. Both algorithms resulted from research in trying to determine the accuracy of the safety factor heuristic of Jacob Savir.