Birnbaum Importance
The Birnbaum importance measure is defined as:
Where:
A indicates that the event whose importance is being measured occurred.
~A indicates that this event did not occur.
X indicates the top event.
The Birnbaum importance measure for the event A is the difference in the probability of the top event given that the event A did occur minus the probability of the top event given that the event A did not occur. This is one measure of the increase in the probability of the top event due to the event A.
Consider a top event X, which is the result of event A and B event being connected by an OR gate. The fault tree would define the top event X to be X={A or B}. Assume that the probability of event A is 0.1 and that of event B is 0.2.
Let P{X|A} denote the probability of the top event X given that the basic event A occurred. Clearly, if A occurs, {A or B} occurs, so that X occurs. Therefore:
P{X|A}=1.0
Also, let P{X|~A} denote the probability of the top event given that the basic event A does not occur. Here, given that A does not occur, X only occurs if the event B occurs. Therefore:
Thus, the Birnbaum importance measure equals:
