Boolean Algebra - Mathematics and Statistics for the Environment - page 212

context of logical reasoning (Boole 1951, 2003). Common practice today is to use Boolean algebra
to solve the logical portion of problems involving probability calculations (Boole 1952, 2003). All of
these applications can be of some benefit to the safety practitioner in his or her goal of reducing acci-
dents. Of particular interest is the use of Boolean techniques in fault-tree analysis (FTA), which is
used in reliability analysis of engineering systems (Clemens and Simmons, 1986; Kolodner, 1971).
Fault-tree analysis is not only an effective combination of probability and backward reasoning but
also Boolean algebra. The probability of some high-level event is the result of a combination of
lower level events. Estimating (or knowing) the probabilities of failure of events in the fault tree
allows one to estimate the probability of success or failure.
Environmental practitioners and students of environmental disciplines are familiar with algebra;
it is a required core subject. Algebra is a logical outgrowth of arithmetic, and many of the methods
of arithmetic are used in algebra, although in modified, expanded, or original form. To a limited
extent, the same relationship can be used in describing Boolean algebra. That is, many of the laws
for Boolean variables are not much different than the laws of numeric algebra. This relationship is
readily seen in the commutative law, distributive law, and less so in addition (identity and inverse
variables) used for real algebra operations.

9.1.1 F ault -t ree a nalysis
In environmental practice, inductive methods of analysis analyze the components of the system and
postulate the effects of their failure on total system performance. Deductive methods of analysis
move from the end event to try to determine the possible causes. They determine how a given end
event could have happened. One widespread application of deductive systems to environmental
health and safety analysis is fault-tree analysis, which postulates the possible failure of a system
and then identifies component states that would contribute to the failure. It reasons backwards from
the undesired event to identify all of the ways in which such an event could occur and, in doing so,
identifies the contributory causes. The lowest levels of a fault tree involve individual components or
processes and their failure modes. This level of analysis generally corresponds to the starting point
in failure mode and effect analysis (FMEA), which is a system reliability analysis that is organized
around the basic question “What if …?”
Fault-tree analysis uses Boolean logic and algebra to represent and quantify the interactions
between events. The primary Boolean operators are AND and OR gates. With an AND gate, the
output of the gate—the event that is at the top of the symbol—occurs only if all of the conditions
below the gate, and feeding into the gate, coexist. With the OR gate, the output event occurs if any
one of the input events occurs.
When the probabilities of initial events or conditions are known, the probabilities of succeed-
ing events can be determined through the application of Boolean algebra. For an AND gate, the
probability of the output event is the intersection of the Boolean probabilities, or the product of the
probabilities of the input events, or:

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

For an OR gate, the probability of the output event is the sum of the “union” of the Boolean prob-
abilities, or the sum of the probabilities of the input events minus all of the products:

Probabilty (output) Prob Input 1) = + (ProbIn put 2) + (ProbInput 3)

(ProbInput 1) × (Prob Input 2)

+ (ProbInput 2) × (ProbInput 3)

−

+ (P robInput 1) × (ProbInput 1) × (ProbInput 3)

+ (ProbInput 1) × (ProbInput 2) × (ProbInput 3)