Probability of Independent Events Not Happening

Theorem
Let $$\mathcal E = \left({\Omega, \Sigma, \Pr}\right)$$ be a probability space.

Let $$A_1, A_2, \ldots, A_m \in \Sigma$$ be independent events in the event space of $$\mathcal E$$.

Then the probability of none of $$A_1$$ to $$A_m$$ occurring is:
 * $$\prod_{i=1}^m \left({1 - \Pr \left({A_i}\right)}\right)$$

Corollary
Let $$A$$ be an event in an event space of an experiment $$\mathcal E$$ whose probability space is $$\left({\Omega, \Sigma, \Pr}\right)$$.

Let $$\Pr \left({A}\right) = p$$.

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 $$\left({1 - p}\right)^m$$.

Proof
Let $$A_1, A_2, \ldots, A_m \in \Sigma$$ be independent events.

From Independent Events are Independent of Complement, we have that $$\Omega \setminus A_1, \Omega \setminus A_2, \ldots, \Omega \setminus A_m \in \Sigma$$ are also independent.

From the definition of occurrence, if $$A$$ does not happen then $$\Omega \setminus A$$ does happen.

So for none of $$A_1, A_2, \ldots, A_m$$ to happen, all of $$\Omega \setminus A_1, \Omega \setminus A_2, \ldots, \Omega \setminus A_m$$ must happen.

From Elementary Properties of Probability Measure:
 * $$\forall A \in \Omega: \Pr \left({\Omega \setminus A}\right) = 1 - \Pr \left({A}\right)$$

So the probability of none of $$A_1$$ to $$A_m$$ occurring is:
 * $$\prod_{i=1}^m \left({1 - \Pr \left({A_i}\right)}\right)$$

Proof of Corollary
It can immediately be seen that this is an instance of the main result with all of $$A_1, A_2, \ldots, A_m$$ being instances of $$A$$.

The result follows directly.