Probability of Event not Occurring

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

Let $\Pr \left({A}\right)$ be the probability of event $A$ occurring.

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

That is, the probability of event $A$ not occurring is $1$ minus the probability of event $A$ occurring.

Proof
From the conditions for $\Pr$ to be a probability measure, we have:


 * $(1): \quad \forall A \in \Sigma: 0 \le \Pr \left({A}\right)$


 * $(2): \quad \Pr \left({\Omega}\right) = 1$


 * $(3): \quad \displaystyle \Pr \left({\bigcup_{i \mathop \ge 1} A_i}\right) = \sum_{i \mathop \ge 1} \Pr \left({A_i}\right)$ where all $A_i$ are pairwise disjoint.

Let $A \in \Sigma$ be an event.

Then $\left({\Omega \setminus A}\right) \in \Sigma$ by definition of Event Space: Axiom $(ES \ 2)$.

From Intersection with Relative Complement, we have that:
 * $A \cap \left({\Omega \setminus A}\right) = \varnothing$

From Union with Relative Complement, we have that:
 * $A \cup \left({\Omega \setminus A}\right) = \Omega$

So:
 * $\Pr \left({A}\right) + \Pr \left({\Omega \setminus A}\right) = 1$

from above, and so:
 * $\Pr \left({\Omega \setminus A}\right) = 1 - \Pr \left({A}\right)$

Also see

 * Elementary Properties of Probability Measure