# 概率

## 历史

Cardano的数学著作中有很多给赌徒的建议。这些建议都写成短文。例如：《谁，在什么时候，应该赌博？》、《为什么亚里士多德谴责赌博？》、《那些教别人赌博的人是否也擅长赌博呢？》等。

## 概念

「從一班40名學生中隨意選出一人，這人會是男生嗎？」

## 數學處理

1. 非负性：$P(A) \geq 0$
2. 规范性：$P( \Omega ) = 1$
3. 可數可加性：对可數个两两互斥事件{Ai}i∈N有：${\displaystyle\sum_{i=1}^{\infty}P(A_{i})=P\left( \bigcup_{i=1}^{\infty}A_{i}\right)}$

### 概率计算总结

A $P(A)\in[0,1]\,$

A或B \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}
A和B \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}
B的情况下A的概率 $P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)} \,$

## 参考文献

### 引用

1. ^ "Probability". Webster's Revised Unabridged Dictionary. G & C Merriam, 1913
2. ^ "Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), ISBN 9780534243128
3. ^ William Feller, "An Introduction to Probability Theory and Its Applications", (Vol 1), 3rd Ed, (1968),Wiley ,ISBN 0-471-25708-7
4. ^ Probability Theory The Britannica website
5. ^ 機率空間 (1)機率論的誕生. 國科會高瞻自然科學教學資源平台. 2011-06-07 [2014-10-21] （中文）.
6. ^ Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
7. ^ Gorman, Michael (2011) "Management Insights". Management Science [完整来源]
8. ^ Olofsson (2005) Page 8.
9. ^ Burgi, Mark (2010) "Interpretations of Negative Probabilities", p. 1. arXiv:1008.1287v1
10. ^ Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. Letter to Max Born, 4 December 1926, in: Einstein/Born Briefwechsel 1916-1955.
11. ^ Moore, W.J. Schrödinger: Life and Thought. Cambridge University Press. 1992: 479. ISBN 0-521-43767-9.

### 书籍

• Olofsson, Peter (2005). Probability, Statistics, and Stochastic Processes, Wiley-Interscience. p. 504. ISBN 0-471-67969-0