Bayes Theorem
If the functional relationships within a group of elements are more complex than simple series or redundant configurations, then the previous guidelines for combining reliabilities may be invalid. In such cases, suitable reliability expressions may often be determined by using derivatives of Bayes Theorem.
The following two examples illustrate the use of the theorem to derive reliability expressions.
Example Power Supply
1. Consider the group of items represented by Reliability Block Diagram 1 below, where D is a power supply common to items 2 and 4 only.
2. For two events, A and B, a formula derived from Bayes Theorem is:
Where:
P(A)= The probability of event A occurring.
P(A/B)= The probability of event A occurring given that event B occurs.
= The probability of event B not occurring.A= The probability of system success.
B= The probability of item D operating successfully.
3. Then, P(A/B) is the probability that the group shown in Reliability Block Diagram 2 below does not fail because this is the system RBD given that D does not fail:
Thus:
4. 
is the probability of the system being successful given that D fails. For this, items 1 and 3 must operate successfully. The system RBD, given that D fails, is thus:
is the probability of the system being successful given that D fails. For this, items 1 and 3 must operate successfully. The system RBD, given that D fails, is thus:
Thus:
5. If the reliability of item D is RD, then:
6. P(A) is equivalent to the system reliability. Thus, substituting the above results in the expression in Step 2:
Example Conditional System Operation
Task
Consider the group of items represented by Reliability Block Diagram 1 below, where the system will operate successfully providing at least one of the following conditions are met:
◦ Items 1 and 3 are operational.
◦ Items 2 and 4 are operational.
◦ Items 1, 5 and 4 are operational.
◦ Items 2, 5 and 3 are operational.
1. Derive the reliability model for the system:
2. Let:
A = System success.
B= Item 5 operating successfully.
Then, following similar procedures to those described in Example 1, the following expressions can be derived:
3. Thus, system reliability, RS, is given by:
