# Probability of Independent Events Not Happening/Corollary

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## Theorem

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

Let $\map \Pr A = p$.

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

Then the probability that $A$ does not occur during the course of $m$ trials of $\EE$ 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.

$\blacksquare$