XOR Gate:
If A1 and A2 are the inputs and A is the output of an XOR gate, then the probability of the gate is:
Pr{A} = Pr{A1 and ~A2} + Pr{A2 and ~A1}
If the events are independent, then:
Example
Consider a fault tree with four basic events: A, B, C and D. The top event is T. The events A and B are connected to an OR gate named Gate1. The events C and D are connected to an XOR gate named Gate2. The gates Gate1 and Gate2 are connected to the top event using an AND gate. Figure 5-4 shows this fault tree.
Assuming that the probabilities of the basic events are:
Pr{A} = 0.1
Pr{B} = 0.2
Pr{C} = 0.3
Pr{D} = 0.5
Then:
Pr{Gate1} = 1 – (1-Pr{A}) (1-Pr{B})
= 1 – (1-0.1) (1-0.2)
= 0.28
Pr{Gate2} = Pr{C} (1- Pr{D}) + (1-Pr{C}) Pr{D}
= 0.3 (1-0.5) + 0.5 (1-0.3)
= 0.5
Pr{top gate} = Pr{T} = Pr{Gate1} Pr{Gate2}
= (0.28) (0.5)
=0.14