概率论快速学习03：概率公理补充

2016-04-28 1452

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简介：

Written In The Font

　　I like maths when i was young,but I need to record them. So I am writing with some demos of Python

Content

　　If two events, A and B are independent then the joint probability is

　　 $P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B),\,$

For example, if two coins are flipped the chance of both being heads is

$\tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{4}.$

In Python

A = set([1,2,3,4,5])
B = set([2,4,3,5,6])
C = set([4,6,7,4,2,1])

print(A & B & C)

Output:

{2, 4}

# & find the objects the same in Set

　　 If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as:

　　 $P(A \cup B)$ .

　　If two events are mutually exclusive then the probability of either occurring is

　　 $P(A\mbox{ or }B) = P(A \cup B)= P(A) + P(B).$

For example, the chance of rolling a 1 or 2 on a six-sided die is

$P(1\mbox{ or }2) = P(1) + P(2) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}.$

In Python

A = set([1,2,3,4,5])
B = set([2,4,3,5,6])
C = set([4,6,7,4,2,1])

print(A | B | C)

Output:

{1, 2, 3, 4, 5, 6, 7}

# | find all the objects the set has

　　If the events are not mutually exclusive then

　　 $P\left(A \hbox{ or } B\right)=P\left(A\right)+P\left(B\right)-P\left(A \mbox{ and } B\right).$

Proved

　　 $\begin{align} P(A\cup B) & =P(A\setminus B)+P(A\cap B)+P(B\setminus A)\\ & =P(A)-P(A\cap B)+P(A\cap B)+P(B)-P(A\cap B)\\ & =P(A)+P(B)-P(A\cap B) \end{align}$

For example:

　　Let’s use Python to show u an example about devil's bones (骰子,不是魔鬼的骨头哈)

A = set([1,2,3,4,5,6])  # the all results of devil's bones
B = set([2,4,3])        # the A event results 
C = set([4,6])          # the B event results 

P_B =  1/2
P_C =  1/3

D = B | C
print(D)

P_D = 2/3

print(P_D == (P_B+P_C - 1/6))

Output:

{2, 3, 4, 6}
True

Let me show u some others :

         $P(A)\in[0,1]\,$
         $P(A^c)=1-P(A)\,$
         $\begin{align} P(A\cup B) & = P(A)+P(B)-P(A\cap B) \\ P(A\cup B) & = P(A)+P(B) \qquad\mbox{if A and B are mutually exclusive} \\ \end{align}$
         $\begin{align} P(A\cap B) & = P(A|B)P(B) = P(B|A)P(A)\\ P(A\cap B) & = P(A)P(B) \qquad\mbox{if A and B are independent}\\ \end{align}$
        $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} \,$

If u r tired , please have a tea , or look far to make u feel better.If u r ok, Go on!

　　Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written:

　　 $P(A \mid B)$ ,

　　Some authors, such as De Finetti, prefer to introduce conditional probability as an axiom of probability:

$P(A \cap B) = P(A|B)P(B)$ ^①

Given two events A and B from the sigma-field of a probability space with P(B) > 0, the conditional probability of A given Bis defined as the quotient of the probability of the joint of events A and B, and the probability of B:　　

　　 $P(A|B) = \frac{P(A \cap B)}{P(B)}$ ^②

　　the ①② expressions are the same. Maybe u can remember one , the other will be easy to be coverted.So I am going to tell an excemple to let u remmeber it(them):

　　“the phone has a power supply (B), the phone can be used to call others(A).”

　　One → $P(A \mid B)$ : When the phone has a full power supply , u can call others.

　　Two →P(B): has a power supply

　　Three = One + Two → U can call others about your love with others.

do u remember it?