Disjoint Fault-Location Events Are Independent

For each $j$, let $X_j=\{0,1{\}}^{L_j}$ be the finite set of fault-indicator patterns on $L_j$, and let $A_j\subseteq X_j$ be the patterns that realize $E_j$. The event $E_j$ is the disjoint union of the pattern events $\{X{\mathrm{\mid }}_{L_j}=x_j\}$ over $x_j\in A_j$.

For any tuple $x_j\in A_j$, the joint pattern event fixes one bit at each location in ${\bigcup }_j{L}_j$. The product fault law [F1] gives its probability as the product of the one-location bit probabilities; this is the finite-intersection condition in [F3].

Summing over the finite Cartesian product $A_1\times \cdots \times A_s$ and distributing finite sums gives $\mathbb{P}({\bigcap }_{j=1}^s{E}_j)={\sum }_{x_1\in A_1,\dots ,x_s\in A_s}{\prod }_{j=1}^s\mathbb{P}(X{\mathrm{\mid }}_{L_j}=x_j)={\prod }_{j=1}^s{\sum }_{x_j\in A_j}\mathbb{P}(X{\mathrm{\mid }}_{L_j}=x_j)={\prod }_{j=1}^s\mathbb{P}(E_j)$. Thus $E_1,\dots ,E_s$ are mutually independent.