F.5 Maths probability
In this question, I try to use a Venn diagram to find the formula in the blue
rectangle. However, i found that I also need to use this formula in another
questions, but I cannot find this formula in the textbook.
I want to ask is this formula is already proved? Is there any other formula about
probability that is very useful but do not mention in textbook?
請問 'set rule' 有沒有中文名稱？
- 知足常樂Lv 75 年 前最佳解答
If events A and B are mutually exclusive, then
P(A or B) = P(A) + P(B).
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.
= P( (A and C) or (A and C') )
= P(A and C) + P(A and C')
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
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：
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: