ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4325 on D1716: Event
Probability one if conditional probability one relative to event of probability one
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) \begin{equation} \mathbb{P}(E) = 1 \end{equation}
(iii) \begin{equation} \mathbb{P}(F \mid E) = 1 \end{equation}
Then \begin{equation} \mathbb{P}(F) = 1 \end{equation}
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) \begin{equation} \mathbb{P}(E) = 1 \end{equation}
(iii) \begin{equation} \mathbb{P}(F \mid E) = 1 \end{equation}
By R2556: Probability calculus expression for probability conditioned on event of nonzero probability, we have \begin{equation} 1 = \mathbb{P}(F \mid E) = \frac{\mathbb{P}(E \cap F)}{\mathbb{P}(E)} = \mathbb{P}(E \cap F) \end{equation} Whence, from R4331: Probability of binary intersection with an almost sure event, we have \begin{equation} 1 = \mathbb{P}(E \cap F) = \mathbb{P}(F) \end{equation} $\square$