This paper treats, in a statistical manner, the class of geophysical observations that sometimes are precursors to a large earthquake, but at other times occur apparently unrelated to any earthquakes in time or space. Each such observation is handled by associating it with the largest ensuing earthquake in a definite time and spatial window; then the probability that the observation is useful is estimated from the difference between this extreme value distribution and the long-term average extreme value distribution for the same region. This probability that the precursor is useful can then be incorporated into calculations of the revised probability of an earthquake following observation of one (a trivial case) or several unreliable precursors to derive the revised probability of a significant earthquake. The model takes full account of the magnitude distribution of earthquakes and of discrete levels of precursor observations. The United States earthquake prediction program is still many years from being able to obtain the full benefits of the model. However, the model suggests useful statistics which the program should gather on unreliable precursors.