Conditional Probability multiplication formula full probability formula and Bayesian Formula

1. Conditional Probability

Define a and B as two events, and P (a)> 0 is called

P (B bought a) = P (AB)/P ()

It is the probability of occurrence of condition Event B Under Condition.

2. Multiplication Formula

Set P (a)> 0

P (AB) = P (B represents a) P ()

3. Full probability formula and Bayesian Formula

Define sample space where S is test E, B1, B2 ,... A group of events whose BN is E. If

1. Bibj =ф, I =j, I, j = 1, 2 ,..., N;

 

B1 then B2 then... ∪ Bn = s

It is called b1, b2 ,..., BN is a division of the sample space.

The Theorem sets the sample space of test E to events where A is E, B1, B2 ,..., A division of BN, and P (BI)> 0 (I = 1, 2 ,... N), then

P (A) = P (A ∣ B1) P (B1) + P (A ∣ B2) +... + P (A ∣ bn) P (BN)

It is called the full probability formula.

The Theorem sets the sample space of the test Russian e as S, A as E event, B1, B2 ,..., A division of BN, then

P (Bi ∣ A) = P (A ∣ Bi) P (BI)/Σ p (B | AJ) P (AJ) = P (B | ai) P (AI) /P (B)

It is called Bayesian formula.

Note: I, j are subscripts, and the sum is 1 to n.