Probability Values and the Fault Tree

TABLE 12.1 Injury Rates for a Sample of the Male and Female Population 

These data indicate that there is a 73% probability that an injury event will involve a male employee. 

when occurrence of the top event is not dependent upon the occurrence of all sub-events, the probability of the top event is not so remote. In this example, the likelihood of simple system failure (as opposed to catastrophic system failure) has increased several orders of magnitude and would therefore require a more informed decision regarding hazard risk acceptance. 

The FTA is a technique which can be used to identify those events which can or must occur in order to realize a desired or undesired outcome. The technique uses a deductive approach to event analysis as it moves from the general to the specific. The FTA has great utility in its ability to distinguish between those events which must occur (represented by an AND gate) and those that simply can occur (represented by an OR gate) in order for the top event to occur. The information charted on a fault tree provides a qualitative analysis by demonstrating how specific events will effect an outcome. If probability data are known for these events, then the FTA can also provide quantitative information to further evaluate the likelihood of achieving the top event. 

Once developed, the fault areas which are responsible for yielding an undesired (or desired) event can be evaluated on the micro rather than the macro level and this is one of the primary utilities of the fault tree analysis technique. 

As discussed in Chapter 5, known or deduced failure rates can assist in developing recommended control actions. In any given cutset, if a failure rate or a combination of failure rates is known to be quite low, then the potential effect of these rates on the top event can be numerically evaluated against the costs of controlling these risks. 

A set of events attached to the main event through an or gate are sometimes referred to as mutually exclusive events since the occurrence of the main event is not dependent on the occurrence of all subevents and occurrence of each subevent is not dependent on the occurrence of the other subevents. This means that, when events in a set are mutually exclusive, the probability of one or another of the events occurring is equal to the sum of the probabilities of the events occurring individually. This old but fundamental concept is known as the addition rule for 

A = first possible event B = second possible event 

As an example, in evaluating the safe behavior of 1000 men and women. Table 12.1 shows the percent of the total that either did or did not have occupational injuries. The contributing events (men having injuries and women having injuries) are not mutually exclusive since either or both may have suffered occupational injuries and, subsequently, affected the primary or top event (safe behavior). Hgure 12.7 shows the fault tree for this extremely simple example. By applying the modified formula for non-mutually exclusive events, the probability of an injury event I involving a man M can be calculated as follows  

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