An indicator function is introduced into Equation 5–4, which adopts the value one if the system, examined for a random parameter set, is in a state considered as failure,and zero else. Formally:

(5–13)

After setting this indicator into the integral over the failure domain, the integration domain can be changed without influencing the value of the integral. The resulting notation suggests that the integral can be interpreted as expected value of the indicator I(g(x)).

(5–14)

The concept of Monte Carlo simulation is simply to generate a sample of random vectors x _i , submit each sample to the simulation of the examined system as input parameter set and compute the limit state function. Then the average of the sample of indicator values is computed as an estimator of the expected value and therefore of the probability of failure.

(5–15)

In the above, x _i is the i ^th realisation in a sample of size m. It can be proven that is an unbiased estimator for the probability of failure.

An estimator which depends on random inputs is a random variable itself. The variance of this estimator is defined as (Rubinstein 1981)

(5–16)

It can be computed approximately from the sample and will approach zero for .

The smaller the variance, the more trustworthy is the estimate. The estimator standard deviation normalized to the sample size,

(5–17)

is also referred to as standard error.

The plain Monte Carlo method is the most general and robust way to compute the probability of failure, yet an inefficient one. The required sample size (and thus the number of solver calls) does not depend on the dimension, but on the expected failure probability. If, for example, a failure probability of shall be computed with a coefficient of variation of 10%, a required sample of size m = 10⁵ is derived from Equation 5–16.