Question

Prove that if s is any sample space and u and v are events in s with u ⊆ v, then p(u) ≤ p(v).

Answer 1

If $U\subseteq V$, then either $U=V$ or $U$ is contained entirely in $V$.

In the first case, we get $\mathbb P(U)=\mathbb P(V)$.

In the second case, we have $\mathbb P(U)=\mathbb P(U\cap V)=\mathbb P(U\mid V)\cdot\mathbb P(V)$, or $\mathbb P(U\mid V)=\dfrac{\mathbb P(U)}{\mathbb P(V)}$. The LHS must be between 0 and 1, and this can only occur as long as $\mathbb P(U)\le\mathbb P(V)$.