Probability of Event not Occurring

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Theorem

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $\map \Pr A$ be the probability of event $A$ occurring.


Then:

$\forall A \in \Sigma: \map \Pr {\Omega \setminus A} = 1 - \map \Pr A$

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


Another way of stating this is:

$\map \Pr A + \map \Pr {\Omega \setminus A} = 1$


Proof

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

$(1): \quad \forall A \in \Sigma: 0 \le \map \Pr A$
$(2): \quad \map \Pr \Omega = 1$
$(3): \quad \displaystyle \map \Pr {\bigcup_{i \mathop \ge 1} A_i} = \sum_{i \mathop \ge 1} \map \Pr {A_i}$ where all $A_i$ are pairwise disjoint.


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

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

From Intersection with Relative Complement is Empty:

$A \cap \paren {\Omega \setminus A} = \O$

From Union with Relative Complement:

$A \cup \paren {\Omega \setminus A} = \Omega$


So:

$\map \Pr A + \map \Pr {\Omega \setminus A} = 1$

from above, and so:

$\map \Pr {\Omega \setminus A} = 1 - \map \Pr A$

$\blacksquare$


Also see


Sources