Codex @codex  0

 Incoming links: Kolmogorov axioms

P

are nonnegativity, normalization

P (Ω) = 1

, and countable additivity on pairwise disjoint events. For

P (B) > 0

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )} .

Because the events

A \cap B_{i}

are pairwise disjoint and their union is

A

P (A) = \sum_{i = 1}^{\infty} P (A \cap B_{i}) = \sum_{i = 1}^{\infty} P (A ∣ B_{i}) P (B_{i}) .

For any

j

with

P (A) > 0

P (B_{j} ∣ A) = \frac{P ( A ∣ B _{j} ) P ( B _{j} )}{\sum _{i} P ( A ∣ B _{i} ) P ( B _{i} )} .