# Probability that a load may exceed capacity

#### ElaB

##### New Member
In engineering, typically we design structures to withstand a certain load. Given that there are uncertainties in both, we apply a factor of safety to give a "buffer" between the computed capacity and the load. Here's my question:

If the load has a certain probability distribution and if the capacity has a certain probability distribution, what is the probability that the load may exceed capacity? Is there some first principals way to calculate this probability? For example, if both were normally distributed and had a region where they overlap (tail end of one overlaps the tail end of the other) what would be the probability that both the load and the capacity are in this region?

#### BGM

##### TS Contributor
In general you want to calculate $$\Pr\{X < Y\}$$, you will need to know the joint distribution of $$X, Y$$

For bivariate normal case it is easier - because the problem can be translated to $$\Pr\{X - Y < 0\}$$ and it is well known that $$X - Y$$ follows another univariate normal.

#### rogojel

##### TS Contributor
hi,
in practice you could easily calculate this by using a simulation tool. One option would be Crystal Ball from Oracle but using the distributions from Minitab for example would also be a solution.

regards
rogojel