概率论高速学习04：概率公理 全概率 贝叶斯 事件独立性

Posted 2020-10-24 llguanli

概率论高速学习04：概率公理 全概率 贝叶斯 事件独立性

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Written In The Font

    数学和生活是技术之本, 有了数学,加上生活,才会开心.

    今天继续概率论:

• 全概率
• 贝叶斯
• 事件独立性

Content

The total probability

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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，当中次品率分别为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随意抽检一件，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

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for some partition {B_j} of the event space, the event space is given or conceptualized in terms of P(B_j) and P(A|B_j). 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

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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 : 

 

Editor\'s Note

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