ThmDex – An index of mathematical definitions, results, and conjectures.
Probability of set difference
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $E \subseteq F$ is a D78: Subset of $F$
Then \begin{equation} \mathbb{P}(F \setminus E) = \mathbb{P}(F) - \mathbb{P}(E) \end{equation}
Subresults
R3719: Probability of complement event
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E, F \in \mathcal{F}$ are each an D1716: Event in $P$
(ii) $E \subseteq F$ is a D78: Subset of $F$
The first claim is a direct corollary to R978: Measure of set difference since the finiteness constraint is automatically satisfied with a probability measure. $\square$