# Probability of Failure

The definition of **Probability of Failure** in **SWedge** (6.0) is:

PF = Number of Failed Wedges / Number of Samples

where:

- Number of Failed Wedges = number of valid wedges with safety factor < Design Factor of Safety
- Number of Samples = number of samples specified in the Project Settings dialog

This definition of the Probability of Failure is considered representative of the overall probability of **wedge failure** for a given set of probabilistic input data.

**Design Factor of Safety**= 1, however, this value can be changed in the Project Settings dialog.

## Probability of Sliding

Another measure of probabilistic failure for wedges is the **Probability of Sliding**. This is defined as:

Probability of Sliding = Number of Failed Wedges / Number of Valid Wedges

where:

Number of Valid Wedges = Number of Samples - Number of Invalid Wedges

The Probability of Sliding is the probability that a kinematically valid wedge will fail (slide).

The generation of **Valid** or **Invalid** wedges is related to the random sampling of orientation distributions (joint, slope or tension crack). Depending on your geometry input, it is possible for the random sampling of the input data to generate invalid wedge geometries.

- A
**Valid Wedge**is a kinematically feasible wedge, which conforms to the wedge plane definitions used in**SWedge**, and which can be removed from the slope. - An
**Invalid Wedge**is a wedge which does NOT conform to the wedge plane definitions used in**SWedge**, or which cannot be removed from the slope. - An Invalid Wedge can also be defined by the settings in the Failure Mode Filter dialog (e.g. if you exclude certain failure modes then these become invalid wedges).

The Number of Valid Wedges is also equal to the Number of Failed Wedges + Number of Stable Wedges.

The Probability of Sliding is listed in the Info Viewer, or you can calculate the value from the probabilistic analysis summary in the Sidebar (i.e. Number of Failed / Number of Valid).

If the statistical sampling of orientations does NOT produce any Invalid Wedges (i.e. Number of Valid = Number of Samples), then the **Probability of Sliding** = **Probability of Failure**.

The Probability of Failure can also be written as:

Probability of Failure = (Probability of Kinematically Valid Wedges) x (Probability of Sliding)

= (Number Valid/Number Samples) x (Number Failed/Number Valid)

= Number Failed / Number Samples

## SWedge Version 4.0 Definition of Probability of Failure

It should be noted that in previous versions of **SWedge** (prior to version 5.0), the Probability of Failure which was reported, was in fact the Probability of Sliding (as defined above).

That is, in **SWedge** version 4.0:

Probability of Failure = Number of Failed Wedges / Number of Valid Wedges

As of **SWedge** version 5.0, the definition of Probability of Failure was revised to the current definition, because the earlier version (where Probability of Sliding was reported as the Probability of Failure), had a tendency to overestimate the Probability of Failure in some cases, which could lead to overly conservative slope designs.

For an interesting discussion of Probability of Failure in rock slopes, click here to read a discussion posted on the Rocscience website.