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[Quite lazy recently, so write harder..... :( ]

A friend just asks me what is Baye's formula.

The simplest form is

P(

*A*|

*B*)P(

*B*)=P(

*B*|

*A*)P(

*A*)

where P(

*A*|

*B*) means the conditional probability of

*A*given that

*B*happens.

Lets we take a frequentist approach. Suppose out of

*N*trials,

*A*appears

*a*times,

*B*appears

*b*times, and in

*c*trials, both

*A*and

*B*occur. Then

P(

*A*|

*B*)=

*c*/

*b*=(

*c*/

*N*)/(

*b*/

*N*)=P(

*A*and

*B*)/P(

*B*)

and therefore

P(

*A*|

*B*)P(

*B*)=P(

*A*and

*B*).

Likewise P(

*B*|

*A*)P(

*A*)=P(

*A*and

*B*). Hence we deduce the Baye's formula.

Example: We throw a dice. Let

*A*={1,2},

*B*={1,3,5},

*C*={2,3,5}.

P(

*A*)=1/3, P(

*B*)=P(

*C*)=1/2, P(

*A*and

*B*)=P({1})=1/6, P(

*A*and

*C*)=1/6, P(

*B*and

*C*)=1/3.

P(

*A*|

*B*)=1/3, P(

*A*|

*C*)=1/3.

P(

*B*|

*A*)=1/2, P(

*B*|

*C*)=2/3.

P(

*C*|

*A*)=1/2, P(

*C*|

*B*)=2/3.

We say that two events

*X*and

*Y*are independent if P(

*X*and

*Y*)=P(

*X*)P(

*Y*). If P(

*Y*)≠0 it is the same as saying that P(

*X*|

*Y*)=P(

*X*).

Now we see that

*B*and

*C*are not independent. However we have

*A*and

*B*are independent, and that

*A*and

*C*are also independent. Not the same kind of independence in our usual sense, right?