Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 33 / 6 / a

Let $E_i=\{M(I_i)=0\}$. For any subset $K\subseteq \{1,\dots ,n\}$, simultaneous voidness is voidness of the union. Since the intervals are disjoint, the assumed void probability functional givesThis includes the empty subset. To check all event patterns explicitly, apply the inclusion-exclusion principle to the complements in a chosen pattern:Thus the void events are mutually independent, and their complementary occupied-cell indicators are mutually independent too. This is joint independence, not merely pairwise independence.