Probability of Independent Events Not Happening/Corollary

Theorem
Let $A$ be an event in an event space of an experiment $\mathcal E$ whose probability space is $\struct {\Omega, \Sigma, \Pr}$.

Let $\map \Pr A = p$.

Suppose that the nature of $\mathcal E$ is that its outcome is independent of previous trials of $\mathcal E$.

Then the probability that $A$ does not occur during the course of $m$ trials of $\mathcal E$ is $\paren {1 - p}^m$.

Proof
This is an instance of Probability of Independent Events Not Happening with all of $A_1, A_2, \ldots, A_m$ being instances of $A$.

The result follows directly.