Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 9 / b

$\Omega$ is the sample space; its elements are elementary outcomes, also called sample points. An event is a measurable subset of $\Omega$, belonging to the sigma-algebra $\mathcal{F}$, rather than necessarily a single outcome.

The missing axiom is normalization: $\mathbb{P}(\Omega )=1$. The zero set function satisfies both displayed requirements but is not a probability measure, showing why this condition is essential.

The incomplete axiom needs countable additivity: for every sequence of pairwise disjoint events $A_1,A_2,\dots \in \mathcal{F}$,The stated two-event version implies finite additivity, but does not supply this requirement for a countably infinite collection. Together with nonnegativity and normalization, this gives the usual probability measure axioms on the sigma-algebra $\mathcal{F}$.

The disjoint decomposition into sets $A=(A\cap B)\cup (A\setminus B)$ uses only events in $\mathcal{F}$. Finite additivity and nonnegativity giveThus $\mathbb{P}(A\cap B)\le \mathbb{P}(A)$ already follows from the two stated requirements. Equality is attained, for example, by choosing $B=A$, since then $A\cap B=A$. More generally it holds whenever $\mathbb{P}(A\setminus B)=0$. Neither normalization nor countable additivity is needed for this particular inequality.