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Axioms of Probability

Definition (Probability Function)

Given sample space SS and event space F\mathcal{F}. A real-valued function PP on F\mathcal{F} is called a probability function if:

  1. P(E)≥0P(E)\geq0
  2. P(S)=1P(S)=1
  3. If EiE_i are disjoint for i∈Ni\in\mathbb{N} then P(⋃i=1∞Ei)=∑i=1∞P(Ei)P(\bigcup_{i=1}^\infty E_i) = \sum_{i=1}^\infty P(E_i)

Proposition (Probability Rules)

Let A1,...,AnA_1,...,A_n be events.

  1. P(∅)=0P(\emptyset)=0
  2. Finite Additivity: If A1,...,AnA_1,...,A_n are disjoint, then: P(⋃i=1nAi)=∑i=1nP(Ai)P(\bigcup_{i=1}^n A_i) = \sum_{i=1}^n P(A_i)
  3. P(A1c)=1−P(A1)P(A^c_1)=1-P(A_1)
  4. If A1⊂A2A_1\subset A_2, then P(A1)≤P(A2)P(A_1)\leq P(A_2)

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Summary of MATH 895