# 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 **addition law of probability**.

## 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:

\(\ds \map \Pr A + \map \Pr B\) | \(=\) | \(\ds \map \Pr {A \setminus B} + 2 \map \Pr {A \cap B} + \map \Pr {B \setminus A}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {\paren {A \setminus B} \cup \paren {A \cap B} \cup \paren {B \setminus A} } + \map \Pr {A \cap B}\) | Set Differences and Intersection form Partition of Union | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {A \cup B} + \map \Pr {A \cap B}\) |

Hence the result.

$\blacksquare$

## Also presented as

Some sources present this result as:

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

## Also known as

This result 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**