[Basics of Probability Theory] Probability | Mathematical Probability | Statistical Probability | Geometric Probability | Three Axioms of Probability Theory

Foreword:

Personal study notes, since they are foreign textbooks, the translated mathematical terms may be slightly different from those in domestic textbooks.

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Foreword:

Definition of 0x00 probability

0x02 Basic Properties of Probability

0x03 axiomatic probability

0x04 geometric probability

Definition of 0x00 probability

probability

In a probability experiment, the degree of possibility of an event occurring A, a real value between 0 and 1, is called probability.

Probability is generally defined in the following three ways:

① Mathematical probability (event probability)

② Statistical probability (event probability)

③ Axiomatic probability

Mathematical probability

It is assumed that all the results in the probability experiment have the same probability, such as $\Omega$ marking the number of sample points in the sample space as $N$ . The number of sample points in A is recorded as $n$ .

$P(A) = \frac{the\, number \, of\, event A \, sample\, points}{the\, number\, of sample\, space\, \Omega \, 's\, sample\, points} = \frac{n}{N}$

Let's call it - the mathematical probability of event A (must be finite)

Statistical probability (law of large number)

A probability experiment is repeated $N$ for times, and the number of $n(A)$ occurrences , and event A is called relative frequency :

$P(A) = \frac{n(A)}{N}$ (Number of occurrences of event A / total number of experiments)

Disadvantage: Statistical probabilities vary with the number of experiments.

law of large numbers

As the number of probability trials increases, statistical probability approaches mathematical probability:

$P(A) = \lim_{N \to \infty }\frac{n(A)}{N}$

Example1：

（1）求1次抛硬幣正面朝上（記為H）的概率。

（2）通過抛10000次硬幣獲得如下結果，在此結果的基礎上計算出正面（H）出現的概率。

          10000次，正面5017次，反面4983次

$Sol:$

（1） $\Omega = \left \{ H,T \right \}, A = {H}$

$P(A) = \frac{n}{N} = \frac{1}{2}$

（2） $P(A) = \frac{n}{N} = \frac{5017}{10000} = 0.5017$

Example2：

投幣三次，是正面的次數 i 為事件 Ai(i=0,1,2,3) 時，求該事件的概率。

$Sol:$

$A_0 = \left \{ TTT \right \}$

$A_1 = \left \{ HTT,THT,TTH \right \}$

$A_2 = \left \{ HHT,HTH,THH \right \}$

$A_3 = \left \{ HHH \right \}$

$P(A_0) = \frac{n(A_0)}{N(\Omega )} = \frac{1}{8}$

$P(A_1) = \frac{n(A_1)}{N(\Omega )} = \frac{3}{8}$

$P(A_2) = \frac{n(A_2)}{N(\Omega )} = \frac{3}{8}$

$P(A_3) = \frac{n(A_3)}{N(\Omega )} = \frac{1}{8}$

Example3：

连续抛两次骰子，第一次抛时出现的点数是3的倍数记为事件A，
第二次抛时出现的点数是3的倍数时记为事件B。
分别求出 P(A) 和 P(A∩B)

$Sol:$

A = { (3,1), (3,2), (3,3), (3,4), (3,5), (3,6),

(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }

B = { (1,3), (1,6), (2,3), (2,6), (3,3), (3,6),

(4,3), (4,6), (5,3), (5,6), (6,3), (6,6) }

A ∩ B = { (3,3), (3,6), (6,3), (6,6) }

$P(A) = \frac{n(A)}{n(\Omega )} = \frac{12}{36} = \frac{1}{3}$

$P(A\cap B) = \frac{n(A\cap B)}{n(\Omega )} = \frac{4}{36} = \frac{1}{9}$

0x02 Basic Properties of Probability

theory:

For the following AB events, the following properties hold:

① $P(\O) = 0$

② $P(A^c) = 1- P(A)$

③ $P(A\cup B) = P(A) + P(B) - P(A\cap B)$

④ $P(A\cup B) \leq P(A) + P(B)$

⑤ If A and B are mutually exclusive, $P(A\cup B) = P(A) + P(B)$

⑥ If $A\subset B$ , $P(B-A) = P(B) - P(A)$

⑥ If $A\subset B$ , $P(A) \leq P(B)$

For the following ABC events, the following properties hold:

0x03 axiomatic probability

When the standard space $\Omega$ defines the aggregation of all events,

If the function defined on the collection

Satisfy the following three axioms :

(A1) For all ,

All our research is based on the premise that probabilities are non-negative

（A2） $P(\Omega ) = 1$

All possible probability is 1

(A3) For pairwise mutually exclusive events

If events are mutually exclusive, the probability that they intersect is the sum of their respective probabilities

$P$ The probability function or probability measure defined above is the probability of event A. $P(A)$

This is called the probability space .

0x04 geometric probability

Throw a particle into a measurable area. If the probability of the cast point falling in any area g in the gate is proportional to the measure of g, and has nothing to do with the position and shape of g, it is called a random experiment. It is a geometric random experiment or a geometric probability, and the measure here is the measure, one-dimensional refers to length, two-dimensional refers to area, three-dimensional refers to volume, etc.

When the specimen space is $\Omega$ a non-additive set, if the length, area, and volume in the standard space are finite and given a geometric measure $m(\Omega )$ , the probability of event A is defined as follows:

At this time , the three major axioms (A1) (A2) (A3) $P$ of the probability function are satisfied .