diff --git a/component_failure_modes_definition/component_failure_modes_definition.tex b/component_failure_modes_definition/component_failure_modes_definition.tex index 0aed33b..0cfefaa 100644 --- a/component_failure_modes_definition/component_failure_modes_definition.tex +++ b/component_failure_modes_definition/component_failure_modes_definition.tex @@ -720,6 +720,22 @@ of interest when analysing systems from a statistical perspective. This is of interest for the application of conditional probability calculations such as Bayes theorem~\cite{probstat}. +Another way to view this is to consider the failure modes of +component, with the $OK$ state, as a universal set $\Omega$, where +all sets within $\Omega$ are partitioned. +Figure \ref{fig:partitioncfm} shows a partitioned set representing +component failure modes $\{ B_1 ... B_8, OK \}$ obeying unitary state conditions. + +\begin{figure}[h] + \centering + \includegraphics[width=350pt,keepaspectratio=true]{./component_failure_modes_definition/partitioncfm.jpg} + % partition.jpg: 510x264 pixel, 72dpi, 17.99x9.31 cm, bb=0 0 510 264 + \caption{Base Component Failure Modes with OK mode partitioned} + \label{fig:partitioncfm} +\end{figure} + + + %%- %%- Need a complete and more complicated UML diagram here diff --git a/component_failure_modes_definition/partitioncfm.dia b/component_failure_modes_definition/partitioncfm.dia new file mode 100644 index 0000000..defacbb Binary files /dev/null and b/component_failure_modes_definition/partitioncfm.dia differ diff --git a/component_failure_modes_definition/partitioncfm.jpg b/component_failure_modes_definition/partitioncfm.jpg new file mode 100644 index 0000000..5ff316d Binary files /dev/null and b/component_failure_modes_definition/partitioncfm.jpg differ diff --git a/survey/survey.tex b/survey/survey.tex index dd6b4a0..4dcc57e 100644 --- a/survey/survey.tex +++ b/survey/survey.tex @@ -665,15 +665,19 @@ the probability that $S_k$ is caused by $B_n$ thus $$ -P(S_k|B_n) = \frac{5.10^{-8} \; 0.1 }{ 10^{-7}} = 0.05 = 5\% +P(S_k|B_n) = \frac{5.10^{-8} .\; 0.1 }{ 10^{-7}} = 0.05 = 5\% $$ -To take an example from the diagram (see figure \ref{fig:partitionbcfm2}), where the base component fault cannot -lead to the system failure $S_k$. Taking say $B_9$ which does not overlap with $S_k$ +Taking an example from the diagram (figure \ref{fig:partitionbcfm2}), where the base component fault cannot +lead to the system failure $S_k$. +Taking say $B_9$ which does not overlap with $S_k$ (i.e. $B_9 \cap S_k = \emptyset $), we can see that $P(S_k | B_9) = 0$. -Bayes theorem applied to $B_9$ becomes $P(S_k|B_9) = \frac{P(B_9) \; 0 }{ 10^{-7}}$ -As this is a factor in the numerator, +Bayes theorem applied to $B_9$ becomes + +$$P(S_k|B_9) = \frac{P(B_9) .\; 0 }{ 10^{-7}} = 0 = 0\%$$ + +As $ P(S_k | B_n)$ is a factor in the numerator, the application of bayes theorem to $B_9$ being a cause for $S_k$ has a probability of zero, as we would expect.