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Assessing risk in engineering, work and life
Assessing risk in engineering, work and life

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Series and parallel

So far, these calculations have looked at the probability of several events all happening. For instance, in the situation described above, a leak would occur only if both detectors failed and the bund failed. In effect, the control systems are working in series (Figure 16a). However, in other instances, an undesired outcome might take place if any one of a number of events occurred. In such a case, the control systems would be working in parallel (Figure 16b). For example, a house could be burgled if either the door was left unlocked or a ground-floor window was left open.

Described image
Figure 16 Controls in series and parallel

Probability of at least one event occurring

The method for calculating the probability of one or more of a number of events occurring is to calculate the probability of none of the events occurring and then subtract this answer from 1.

For the special case of either or both of two events occurring (A or B), the probability is given by

multiline equation row 1 probability equals probability of plus probability of minus probability of row 2 cap a occurring cap b occurring both occurring full stop

In probability notation, this is expressed as

equation left hand side cap p times open cap a union cap b close equals right hand side cap p of cap a plus cap p of cap b minus cap p times open cap a intersection cap b close comma

where the symbol union denotes OR.

Note that the probability of both events occurring has to be deducted to avoid double counting. When the probability of cap a occurring is calculated, those occasions when cap a and cap b both occur are also included. Similarly, when the probability of cap b occurring is calculated, the probability of both cap b and cap a occurring is included. Deducting one of these eliminates this double counting, as shown in the probability space diagram in Figure 17.

Described image
Figure 17 Probability space diagram for cap p times open cap a union cap b close