EX 2.13 components in series

Consider a system of five identical components connected in series, as illustrated in Figure 2.3.

Figure 2.3 A system of five components connected in a series

Denote

• a component that fails by $F$
• one that doesn’t fail by $S$ (for success).

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Let $A$ be the event that the system fails. For $A$ to occur, at least one of the individual components must fail.

Outcomes in $A$ include $SSFSS$ ( 1, 2, 4, and 5 all work,but 3 does not), $FFSSS$, and so on. There are in fact 31 different outcomes in $A$. However, $A\prime$, the event that the system works, consists of the single outcome $SSSSS$. We will see in Section 2.5 that if $90$ of all such components do not fail and different components fail independently of one another, then $P\left(A\prime \right)=P\left(SSSSS\right)={.9}^5=.59$. Thus $P\left(A\right)=1-.59=.41$; so among a large number of such systems, roughly $41\%$ will fail.

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