property is not greater than 1

In general, the foregoing proposition is useful when the event of interest can be expressed as “at least …,” since then the complement “less than …” may be easier to work with (in some problems, “more than …” is easier to deal with than “at most …”). When you are having difficulty calculating $P\left(A\right)$ directly, think of determining $P\left(A^{\prime }\right)$.

Property

For any event $A,P\left(A\right)\le 1$.

\begin{proof} This is because $1=P\left(A\right)+P\left(A^{\prime }\right)\ge P\left(A\right)$ since $P\left(A^{\prime }\right)\ge 0$. \end{proof}