  We report an exact analysis of a discrete form of the Chakrabarti-Stinchcombe
model for earthquakes [Physica A \textbf{270}, 27 (1999)] which considers a
pairof dynamically overlapping finite generations of the Cantor set as a
prototype of geological faults. In this model the $n$-th generation of the
Cantor set shifts on its replica in discrete steps of the length of a line
segment in that generation and periodic boundary conditions are assumed. We
determine the general form of time sequences for the constant magnitude
overlaps and hence obtain the complete time-series of overlaps by the
superposition of these sequences for all overlap magnitudes. From the
time-series we derive the exact frequency distribution of the overlap
magnitudes. The corresponding probability distribution of the logarithm of
overlap magnitudes for the $n$-th generation is found to assume the form of the
binomial distribution for $n$ Bernoulli trials with probability 1/3 for the
success of each trial. For an arbitrary pair of consecutive overlaps in the
time-series where the magnitude of the earlier overlap is known, we find that
the magnitude of the later overlap can be determined with a definite
probability; the conditional probability for each possible magnitude of the
later overlap follows the binomial distribution for $k$ Bernoulli trials with
probability 1/2 for the success of each trial and the number $k$ is determined
by the magnitude of the earlier overlap. Though this model does not produce the
Gutenberg-Richter law for earthquakes, our results indicate that the fractal
structure of faults admits a probabilistic prediction of earthquake magnitudes.
