Examples

Reliability: Series, parallel, series

Suppose a process has internal components arranged like this:

center

Write $W_{i}$ for the event that component $i$ succeeds, and $W_{i}^{c}$ for the event that it fails.

The success probabilities for each component are given in the chart:

1	2	3	4	5
92%	89%	95%	86%	91%

Find the probability that the entire system succeeds.

Solution

For intersections, use $P [A B] = P [A] P [B]$ (independence) and for unions, use $P [A \cup B] = 1 - P [A^{c} B^{c}]$ .

So $P [{(W_{2} W_{3})}^{c}] = 1 - (.89) (.95)$ and:

\begin{matrix} success = W_{1} (W_{2} W_{3} \cup W_{4} \cup W_{5}) \\ ≫ ≫ P [success] = (.92) (1 - (1 - (.89) (.95)) (.14) (.09)) ≫ ≫ \approx 0.918 \end{matrix}