= Solution
The missing axiom is \b[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 \b[<countable additivity>]: for every sequence of pairwise disjoint <events> $A_1,A_2,\ldots\in\mathcal F$,
$$
\boxed{\mathbb P\!\left(\bigcup_{j=1}^\infty A_j\right)=\sum_{j=1}^\infty\mathbb P(A_j).}
$$
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> $\mathcal F$.
Back to article page
