Addition Law of Probability/Proof 1

Theorem

Let $\Pr$ be a probability measure on an event space $\Sigma$.

Let $A, B \in \Sigma$.


Then:

$\map \Pr {A \cup B} = \map \Pr A + \map \Pr B - \map \Pr {A \cap B}$

That is, the probability of either event occurring equals the sum of their individual probabilities less the probability of them both occurring.


This is known as the addition law of probability.


Proof

By definition, a probability measure is a measure.

Hence, again by definition, it is a countably additive function.

By Measure is Finitely Additive Function, we have that $\Pr$ is an additive function.

So Additive Function is Strongly Additive can be applied directly.

$\blacksquare$

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