In the system level, adaptive fault diagnosis problem we must determine which components (chips) in a system are defective, assuming the majority of them are good. Chips are tested as follows: Take two chips, say x and y, and have x report whether y is good or bad. If x is good, the answer is correct, but if x is bad, the answer is unreliable and can be either wrong with probability α or right with probability 1 − α. The key to identifying all defective chips is to identify a single good chip which can then be used to diagnose the other chips; the chip problem is to identify a single good chip. In [1] we have shown that the chip problem is closely related to a modified majority problem in the worst case and have used this fact to obtain upper and lower bounds on algorithms for the chip problem. In this paper, we show the limits of this relationship by showing that algorithms for the chip problem can violate lower bounds on average performance for the modified majority problem and we give an algorithm for the “biased chip” problem (a chip is bad with probability p) whose average performance is better than the average cost of the best algorithm for the biased majority problem.