F.5 Maths probability

Probability Rule:
If events A and B are mutually exclusive, then
P(A or B) = P(A) + P(B).

Set Rule:
For any sets (events) A and C, A can be partitioned into two parts, one is with C, one is without C. That is,
A = ( (A and C) or (A and C') )

Clearly (A and C) and (A and C') are two mutually exclusive events.
There is no overlapping part in the Venn diagram.

Therefore,
P(A)
= P( (A and C) or (A and C') )
= P(A and C) + P(A and C')

Other formulae:
P(A or B) = P(A) + P(B) - P(A and B)　[must be true]P(A or B) = P(A) + P(B)　[if A and B are mutually exclusive, i.e., P(A and B) = 0]P(A and B) = P(A | B)P(B) = P(B | A)P(A)　[must be true]P(A and B) = P(A)P(B)　[if A and B are independent]Actually the set rule mentioned above is not strange to you, you also encounter this in the tree diagram.Suppose your tree is:　　　　／　Ｂ
　　　／
　Ａ　－－　Ｂ’
／
＼
　Ａ’　－－　Ｂ
　　　＼
　　　　＼　Ｂ’

Then, you also know that
P(B) = P(A)P(B|A) + P(A')P(B|A')
That is actually just
P(B) = P(B and A) + P(B and A')


2015-03-29 15:13:23 補充：
我以上打的 Set Rule 不是專有名詞，只是一句普通的英文，指 some rules about the concept of "set theory".

如果你對 set theory 有興趣可以看看：
https://mathtrielhighschool.files.wordpress.com/2011/09/18ozxhywnr.pdf

由 P.31 起有很多 rules，不是太難明。

其實把 A 寫成 (A and B) or (A and B') 只是很直接的 partition on B，不是太複雜。

2015-03-29 15:13:43 補充：
如果你真的要一個明確的 rule，這個叫 law of total probabilty:

http://en.wikipedia.org/wiki/Law_of_total_probability