Fig. 4. This illustrates the calculation of probabilities in a more complex FTA section.
may be referred to as intermediate events. Te “top event” is the result of all intermediate and basic events.
Calculating Probability
While the effort required to estab- lish a sequence of causes and effects in logical interconnection is instructive and valuable in itself, the great power of the tool lies in its ability to quantify the probability of the “top event” and to assess the contribution of the basic and intermediate events in numerical terms. If one were to establish the probability of occurrence of the basic events, then by using the mathematics of probability of independent events, a probability can be calculated for the “top event.” For two independent events that
Table 1. Contribution of Causal Branches Top Event Probability:
Event A Event B Event C
Calculated Probability 0.000031
0.000003 0.000048
Table 2. Contribution Analysis of Event A Event A Probability:
Calculated Probability
Event D Event E
0.00001 0.000021
36 | MODERN CASTING September 2014 0.000031
Percent Contribution to Event A
32.3% 67.7%
Percent Contribution to the Top Event
12.2% 25.6%
Fig. 5. This FTA is used to illustrate contribution analysis. The “top event” is caused by intermediate events A, B and C, which in turn are caused by other events. The circles indicate basic events. Probabili- ties of occurrence are provided next to the event.
must occur together, the probability of the result is the product of the events necessary for its occurrence. When events are connected by an “AND,” the “top event” probability is given as P1 x P2 = P. Where three events are so related (and independent), each having probabilities P1, P2 and P3, then the “top event” likelihood would be given as P1 x P2 x P3 = P. Figure 3 illustrates the calculation
of probability when two events are connected by an “OR” relationship. Where either one of two independent events can cause the “top event,” the probabilities P1 and P2 combine as:
0.000082
Percent Contribution to the Top Event
37% 3.7%
58.5%
P = P1 + P2 – P1P2 Te subtracted portion represents
that overlapping probability where both events occur at the same time. It is not correct to double count these occurrences. Where three events are related by an “OR” connection: P = P1 + P2 + P3 – P1P2 – P2P3 – P1P3 + P1P2P3 To summarize the math, the prob-
abilities for “AND” connections are multiplied and the probabilities for “OR” logical connections are added. Figure 4 provides an illustration of a section of an FTA where the calcula- tion rules are applied.
Assigning Probability Te basic events in such chains of
cause and effect generally will fall into three categories, each with its own source of probability information. Equipment Failure: Many basic
events can be the failure or fault of a piece of equipment or a component within a larger system of equipment, such as a plumbing leak or an electri- cal failure. Establishing a frequency for these events relies on maintenance records. Even if maintenance does not keep such records directly, the purchasing department may have records for the purchase of spare parts
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