Probability of Set Difference of Events

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

Let $A, B \in \Sigma$ be events of $\EE$.

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

Then:
 * $\map \Pr {A \setminus B} = \map \Pr A - \map \Pr {A \cap B}$

Proof
From Set Difference and Intersection form Partition:


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

So, by the definition of probability measure:
 * $\map \Pr A = \map \Pr {A \setminus B} + \map \Pr {A \cap B}$

Hence the result.