Bayes' theorem

conditional probability

A/B are event, when B happen, A‘s conditional probability. ps.P(B) > 0

$$P(A|B) = \frac{P(AB)}{P(B)}$$

seven ball in p1 p2
two white ball in p1
one white ball in p2
What’s the probability when fetch white from B bucket.

Event A is fetch white
Event B is fetch from P2 bucket

$$P(white|bucketB) = \frac{P(white and bucketB)}{P(bucketB)} = \frac{\frac{1}{7}}{\frac{3}{7}}=\frac{1}{3}$$

Multiplication formula

1. formula

   $$P(AB) = P(A|B)P(B) = P(B|A)P(A)$$

2. when n >= 2 :

   $$P(A_{1}A_{2}...A_{n-1}A_{n})=P(A_{1})P(A_{2}|A_{1})P(A_{3}|A_{1}A_{2})...P(A_{n}|A_{1}A_{2}...A_{n-1})$$

3. full probability formula
   if event array B1, B2, B3, … satisfying

• $$B_{i} \cap B_{j} = \varnothing$$
• i ≠ j
• i,j=1 2 ….
• $$P(B_{i})>0,i=1,2,....$$
• $$B_{1} \cap B_{2} \cap .... = \Omega$$

then array event is a divide of sample space Ω .

$$P(A) = \displaystyle\sum_{i=1}^nP(B_{i})P(A|B_{i})$$

this formula mean no interference event array can calculate target sample space in divide.
It’s called prior probability

For example: spot check a set of different products’ good rate

bayes formula

bayes formula is the reverse process of full probability formula.
It research a happened event cause factor, not forcast it.
It’s called posterior probability

$$P(B_{i}|A) = \frac{P(B_{i})P(A|B_{i})}{\displaystyle\sum_{j=1}^jP(B_{j})P(A|B_{j})}$$

For example:
find killer in criminal suspect
spam filtering in e-mail system

https://www.cnblogs.com/ohshit/p/5629581.html