第一章关于概率

概率的三大性质：

P(A) >= 0

P(全集) = 1

For A1....An disjoint events, P(A1 U .... U An) = P(A1) + ... + P(An)

条件概率：

已知B发生，A的发生概率：就是这块面积

P(A|B) = P(A并B)/P(B) 

以此也可以得到 P(A，B) = P(A|B) * P(B)

著名的贝叶斯公式：

P(B|A) = P(A|Bi) * P(Bi) / P(A)

往往这里的P(A)需要用全概率公式获得

举例：Have 2 white and 2 balck balls in urn, draw 2 balls without replacement, what's probability both balls are black?

P(both black) = P(b1 is black, b2 is black) = P(b1 is black) * P(b2 is black | b1 is black) = 2/4 * 1/3 = 1/6

全概率公式：

注意这里的A1....An必须是全集的一个完备事件组，什么是完备时间组呢？就是说任意两个Ai, Aj都是disjoint的并且所有的A1 U ....U An是全集！

P(B) =\sigma P(Ai) P(B|Ai)

举例：

have 4 white, 3 black, 2 red socks in drawer, what is prob of drawing a match pair?

A1 = 1st sock is white

A2 = 1st sock is black

A3 = 1st sock is red

B = matched pair

partition of 全集！！也就是说A1, A2, A3不可能同时发生但是也不得不发生一个

P(B) = P(A1) * P(B|A1) + P(A2) * P(B|A2) +P(A3) * P(B|A3) = 4/9 * 3/8 + 3/9 * 2/8 + 2/9 * 1/8 = 20/72