Probability

P(event X)= (number of outcomes consists the event X) / (number of all possible outcomes)
P(A|B) means "under the probability of the event B happened, to find the probability of event A is also happened."As P(B) is the probability of the event B happened, and,if they are dependent event, then, P(A∩B) is the probability of the event A and B happened. So,P(A|B) = P(A∩B) / P(B)

If event A and event B are independent event, then P(A∩B) = P(A) * P(B)So, in this case, P(A|B) = P(A) * P(B) / P(B) = P(A)
(provided that there is a probability that event B will be happened)

Definition 同 axiom 都唔使 prove.
你唔 define 啲公理，何來證明其他啲定理呢！

Agree with 知足常樂，there is no way to proof a definition.
In many sense, that's how mathematics works, see this joke

http://www.netfunny.com/rhf/jokes/89q2/sohrt.425.html

Be careful with P(B) = 0, this is often omitted - but when P(B) = 0, the conditional probability is undefined.

Definition (定義) 無得問點解。

你應該係問錯問題。

2015-03-23 22:20:38 補充：
如果有網友可以答得你滿意的話, 相信只係說明一些相關的資料, 並不是回答你問的 "why"。

2015-03-24 01:17:00 補充：
Definition 是沒有得 prove 的。

P(A | B) is called a conditional probability of an event A given the occurrence of an event B, this is defined as P(A and B)/P(B).

The above definition is valid as long as B is possible, that is, P(B) > 0.
It is okay no matter A and B are dependent or independent.

2015-03-24 01:17:46 補充：
However, if A and B are independent (which means P(A and B) = P(A)P(B)), then
P(A|B)
= P(A and B)/P(B)
= P(A)P(B)/P(B)
= P(A)