In the redundancy combination shown in Figure 7.5-7, Unit A has
two parallel redundant elements, Unit B has three parallel redundant elements,
and Unit C has only one element. Assume that all elements are independent. For
Unit A to be successful, A_{1} or A_{2} must operate; for Unit
B success, B_{1} , B_{2} or B_{3} must operate; and C
must always be operating for block success. Translated into probability terms,
the reliability of Figure 7.5-7 becomes:

R
= [ 1 - P( _{1})
• P(_{2})] • [1 - P(_{1}) • P(_{2}) • P(_{3})] • P(C)

If the probability of success, p, is the same for each element
in a unit,

R
= [1 - (1 - p_{A})^{2 }] • [1 - (1 -
p_{B})^{3 }] • p_{c}

= (1 - q_{A}^{2 })
• (1 - q_{B}^{3} ) • pc

where:

q_{i}
= 1 - p_{i}

Often there is a combination of series and parallel redundancy
in a block as shown in Figure 7.5-8. This arrangement can be converted into
the simple parallel form shown in Figure 7.5-8 by first evaluating the series
reliability of each path:

p_{A}
= p_{a1
}p_{a2}

p_{B }
= p_{b1}
p_{b2} p_{b3}

where the terms on the right hand side represent element
reliability. Then block reliability can be found from:

R
= 1 - (1 - p_{A} ) • (1 - p_{B} )

=
1 - q_{A} q_{B}