Let’s consider a performance function of system.
which describes the functional relationships among the random
numbers. The failure surface or the limit state of interest can then be defined
as 
. This is the boundary
between the safe and failure regions in the design variable space. Suppose that
and 
are the two basic random variables,
the failure surface and the safe and failure regions are shown in Figure 1.
Figure 1 Limit state concept
The Monte Carlo simulation generates the random
samples of the variables according to their PDFs and PMFs, which
is described in the previous section. Then, the random response 
can be evaluated from the performance
function 
. Suppose that
failure region is defined as 
shown in the figure. Let 
be the number of sample points
when and 
be the total number of sample points.
Therefore, an estimate of the probability of failure can be expressed as
