This article is all about the basics of probability. There are two interpretations of a probability, but the difference only matters when we will consider inference.

  • Frequency
  • The degree of belief

Axioms of Probability

A function \(P\) which assigns a value \(P(A)\) to every event \(A\) is a probability measure or probability distribution if it satisfies the following three axioms.

  1. \(P(A) \geq 0 \text{ } \forall \text{ } A\)
  2. \(P(\Omega) = 1\)
  3. If \(A_1, A_2, …\) are disjoint then \(P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i) \)

These axioms give rise to the following five properties.

  1. \(P(\emptyset) = 0\)
  2. \(A \subset B \Rightarrow P(A) \leq P(B)\)
  3. \(0 \leq P(A) \leq 1\)
  4. \(P(A^\mathsf{c}) = 1 – P(A)\)
  5. \(A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)\)

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