概率论快速学习04：概率公理 全概率 贝叶斯 事件独立性
The total probability
In the Set :
The law of total probability is the proposition that if
is a finite or countably infinitepartition of a sample space (in other
words, a set of pairwise disjoint events whose union is the entire
sample space) and each event
is measurable, then for any event
of the same probability space:
example:
例. 甲、乙两家工厂生产某型号车床，其中次品率分别为20%, 5%。已知每月甲厂生产的数量是乙厂的两倍，现从一个月的产品中任意抽检一件，求该件产品为合格的概率？
设A表示产品合格，B表示产品来自甲厂
Bayes
for some partition {Bj} of the event space, the event space is given or conceptualized in terms of P(Bj) and P(A|Bj). It is then useful to compute P(A) using the law of total probability:
example:
An entomologist spots what might be a rare subspecies of beetle, due to the pattern on its back. In the rare subspecies, 98% have the pattern, or P(Pattern|Rare) = 98%. In the common subspecies, 5% have the pattern. The rare subspecies accounts for only 0.1% of the population. How likely is the beetle having the pattern to be rare, or what is P(Rare|Pattern)?
From the extended form of Bayes' theorem (since any beetle can be only rare or common),
One more example:
Independence
Two events
Two events A and B are independent if and only if their joint probability equals the product of their probabilities:
-
.
Why this defines independence is made clear by rewriting with conditional probabilities:
how about Three events
sometimes , we will see the Opposition that can be used to make the mess done. We will use the rule of independence such as :