Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 2 / 6

The inclusion-exclusion principle for probabilities givesEach outcome belonging to two events is initially counted twice and must lose one count. An outcome in all three is counted three times and then subtracted three times, so the last term restores its single count.

Write $U=A\cup B\cup C$. Since $D\subseteq U$, their intersection is $U\cap D=D$. Therefore the definition of conditional probability givesThe positive-probability condition is necessary: if $P(D)=0$, this elementary definition of conditional probability does not assign a value.