Probability Space

A probability space $(\Omega, \mathcal{F}, P)$ models an experiment through the following,

  1. Sample Space. $\Omega$ is the set of all outcomes from the experiment.
  2. Event Space. $\mathcal{F}$ is the set of all allowable **events, subsets of $\Omega$.
  3. Probability Law. $P$ is a measure $\mathcal{F} \rightarrow [0, 1]$.

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The sample space of an experiment is constructed to be collectively exhaustive (i.e. all outcomes of the experiment are accounted for) and mutually exclusive (i.e. such that no two elements in the sample space correspond to the same outcome).

Axioms of Probability

The following axioms govern the behavior of the probability law $P$.

$$ \textbf{Non-negativity}. \hspace{15pt} \forall A \in \mathcal{F}, P (A) \geq 0 \hspace{148pt} \\ \textbf{Additivity}. \hspace{15pt} \forall A_1, A_2, \dots \in \mathcal{F}, P(A_1 \cup A_2 \cup \dots ) = P(A_1 ) + P(A_2) + \dots \\ \textbf{Normalization}. \hspace{15pt} P(\Omega) =1 \hspace{190pt} $$

The following facts can be proven from these axioms.

$$ A \sube B \implies P(A) \leq P(B) \hspace{115pt} \\ P(A \cup B) = P(A) + P(B) - P(A \cap B) \hspace{64pt} \\ P(A^c) = 1 - P(A) \hspace{154pt} \\ P(A \cup B \cup C) = P(A) + P(A^c \cap B) + P(A^c \cap B^c \cap C) $$