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)=(\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)`.