The accuracy of the estimate of 
may depend on the number of sample
points(
). A small number of
sample points give the estimate of subject to considerable error. The
estimate of probability failure would approach the true value as approaches infinity. Thus, the
accuracy of the estimation has been studied in several ways.
Variance of Estimation
One way would be to evaluate the variance or COV of
the estimated probability of failure. The COV can be estimated by
assuming each simulation to constitute a Bernoulli trial, and the number of
failure(
) in trials can be considered to follow a
binomial distribution. Then, the COV of the estimate can be defined as
When you compare the results of Monte Carlo
Simulation, a smaller value of 
is more accurate result of them. The
above equation represents that COV approaches zero as approaches infinity.
Confidence Interval of Estimation
Another way to study the error associated with the number of
simulations is by approximating the binomial distribution with a normal
distribution and estimating the 
confidence interval of the estimated
probability of failure. It can be defined as
where 
is
the exact probability of failure. The percentage error of the probability of
failure can be defined as
Combining those two equations, the percentage error can be simplified as
For example, for 10% error with 95% confidence, the required number of samples is
If is 0.01, then the required is 39,204 for 10% error with
95% confidence. Table 1 lists the required samples for 10% error with 95%
confidence interval.
|
Prob |
0.01 |
0.02 |
0.03 |
0.04 |
0.05 |
0.1 |
0.2 |
0.3 |
0.4 |
0.5 |
|
N |
39204 |
19404 |
12804 |
9504 |
7524 |
3564 |
1584 |
924 |
594 |
396 |
Table 1 The required samples for exact probabilities (for 10% error with 95% confidence)