Addition Law of Probability

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 .

Proof 1

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.

$\blacksquare$

Proof 2

From Set Difference and Intersection form Partition:

$A$ is the union of the two disjoint sets $A \setminus B$ and $A \cap B$

$B$ is the union of the two disjoint sets $B \setminus A$ and $A \cap B$.

So, by the definition of probability measure:

$\map \Pr A = \map \Pr {A \setminus B} + \map \Pr {A \cap B}$

$\map \Pr B = \map \Pr {B \setminus A} + \map \Pr {A \cap B}$

From Set Difference is Disjoint with Reverse:

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

Hence:

Hence the result.

$\blacksquare$

Also presented as

Some sources present the in the form:

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

Also known as

The is also known as the sum rule, but then so are other results in mathematics.

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): addition law
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): addition law (probability)