A fundamental axiom of probability is that if we have two events $A$ and $B$ from the same probability space, and $A$ is a subset of $B$ ($A\subseteq B)$, then $\mathbb{P}(A)\le \mathbb{P}(B)$. This should be intuitive: $A$ occurring is one of the ways that $B$ can occur, but there are also other ways $B$ can happen without $A$, so its probability must be higher. This axiom is known as the monotonicity of probability measures; in this course, for the sake of brevity I typically refer to this as the principle of inclusion.

A great deal of statistical theory involves inequalities, and a typical scenario in which this arises is: we know that $b<c$, therefore $\mathbb{P}(a<b)\le \mathbb{P}(a<c)$. This is intuitively simple, but it is easy to make mistakes with respect to the sign. As a mnemonic device, I suggest “if large side gets larger, or the small side gets smaller, the probability goes up”, but use whatever works for you.

As a more specific example, suppose $X$, $Y$, and $Z$ are random variables. Then by combining the triangle inequality and the monotonicity of probability measures, we have