Probability of Independent Events Not Happening

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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:

$\displaystyle \prod_{i=1}^m \left({1 - \Pr \left({A_i}\right)}\right)$

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

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.

$\forall A \in \Omega: \Pr \left({\Omega \setminus A}\right) = 1 - \Pr \left({A}\right)$

So the probability of none of $A_1$ to $A_m$ occurring is:

$\displaystyle \prod_{i=1}^m \left({1 - \Pr \left({A_i}\right)}\right)$

$\blacksquare$

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.

$\blacksquare$