YTC 發問於 科學及數學數學 · 6 年前

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?

Thank you!

RE 知足常樂：

1 個解答

• 6 年前
最愛解答

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/20...

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_probabil...