Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Probability Theory (Definition)

The branch of mathematics concerning numerical descriptions of how likely an event is to occur.
📜

The statement of the theorem

A probability theory is formally defined by a probability space (Ω,F,P)(\Omega, \mathcal{F}, P), where: \n\n1. Ω\Omega is the sample space, representing the set of all possible outcomes. \n2. F\mathcal{F} is a σ\sigma-algebra on Ω\Omega, meaning F\mathcal{F} is a subset of P(Ω)\mathcal{P}(\Omega) (the power set of Ω\Omega) such that: \n a) Ω∈F\Omega \in \mathcal{F} (The sample space is an event). \n b) If A∈FA \in \mathcal{F}, then its complement Ac=Ω∖AA^c = \Omega \setminus A is also in F\mathcal{F} (Closure under complementation). \n c) If (Ai)i=1∞(A_i)_{i=1}^{\infty} is a countable sequence of sets in F\mathcal{F}, then their union ∪i=1∞Ai\cup_{i=1}^{\infty} A_i is also in F\mathcal{F} (Closure under countable unions). \n3. PP is the probability measure, a function P:F→[0,1]P: \mathcal{F} \to [0, 1], satisfying the Kolmogorov axioms:\n a) Non-negativity: P(A)≥0P(A) \ge 0 for all A∈FA \in \mathcal{F}. \n b) Normalization: P(Ω)=1P(\Omega) = 1. \n c) Countable Additivity: For any countable sequence of pairwise disjoint events (Ai)i=1∞(A_i)_{i=1}^{\infty} in F\mathcal{F}, the probability of their union is the sum of their individual probabilities:\n P(∪i=1∞Ai)=∑i=1∞P(Ai)P(\cup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)