Boolean Algebra - Mathematics and Statistics for the Environment - page 213

Room
Dark

(OR gate)

Both lights
burned out

Power off

(OR gate) (AND gate)

Light 2
burned out

Power
supply failed

Light 1
burned out

Fuse blown

Switch
opened

FIGURE 9.1 Fault tree analysis of light bulbs.

Where the probabilities of the input events are small (less than 0.1, for example), the probability
of the output event for an OR gate can be estimated by the sum of the probabilities of the input
events, or

Probability (output) = (Prob Input 1) + (Prob Input 2) + (Prob Input 3)

Figure 9.1 illustrates a simple FTA. The basic events in Figure 9.1 are represented as rectangles
rather than circles because they could possibly be developed (reduced) further. Assume, for exam-
ple, that in Figure 9.1 the following probabilities exist:

Probability of power supply failing = 0.0010
Probability of switch open = 0.0030
Probability of blown fuse = 0.0020
Probability of light 1 out = 0.0300
Probability of light 2 out = 0.0400

Because the power will be off if the power supply fails or the switch is open or the fuse is blown,
the probability of the power being off is the sum of the probabilities of the power supply failing, the
switch being open, and the fuse being blown, or 0.0010 + 0.0030 + 0.0020 = 0.0060. Both lights will
be burned out if light 1 is out and light 2 is out. The probability of both lights being out is 0.0300
× 0.0400, or 0.0012. The probability of the room being dark because the power is off or both lights
are burned out is 0.0060 + 0.0012, or 0.0072.
The concepts of cut sets and path sets are useful in the analysis of fault trees. A cut set is any
group of contributing elements that if all occur will cause the end event to occur. A path set is any
grouped of contributing elements that if none occurs will prevent the occurrence of the end event.
For the example in Figure 9.1, the end event will occur if