A general approach to Bell-type inequalities for trichotomic observabl
es is presented. With this formalism, which explicitly includes no-cou
nts events, we show that the limits of all Bell-type inequalities are
functions of the supplementary assumptions. This approach enables us t
o include in the causal and local theory the same assumptions used in
the quantum-mechanical analysis of an experiment, and to deduce the in
equality that must be satisfied by any local realistic model incorpora
ting these assumptions. We prove that, using the quantum-mechanical as
sumption of ''random nondetection'' in the causal and local approach,
a generalized Bell-type inequality is deduced whose upper limit can be
violated by the quantum-mechanical predictions for symmetrical experi
mental setups involving detectors with quantum efficiency greater than
0.811.