[Basics of Probability Theory] Conditional Probability | Multiplication Rule

Foreword:

Personal review notes, because they are foreign textbooks, so the translated mathematical terms may be slightly different from domestic textbooks.

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Ⅰ. Conditional probability (condition probability)

0x00 Definition

0x01 Supplement

Ⅱ. Multiplication Law

0x00 official

0x01 proof

Ⅲ. Independence of events

0x00 independent (independent)

0x01 independent and dependent

0x02 pairwisely independent

Ⅰ. Conditional probability (condition probability)

0x00 Definition

In the probability event ( $\Omega,$ $,P$ ), $A,B\in$ and $P(A) > 0$ when,

$A$ The probability of an event occurring under the premise that the event has occurred $B$ : $P(B|A) = \frac{P(A\cup B)}{P(A)}$ , we call it conditional probability.

0x01 Supplement

Function defined on a collection : $P(\cdot |A):$ $\rightarrow \mathbb{R}$

$B\mapsto P(B|A)$ , is the probability function.

(A1) For all $B\in$ , $0\leq P(B|A) < 1$

（A2） $P(\Omega |A) = 1$

(A3) Pairwise mutually exclusive events:

$A1,A2...,P(\bigcup_{i=1}^{\infty }A_i|A) = \sum_{i=1}^{\infty }P(A_i|A)$ Then ( $\Omega,$ $,P(\cdot |A)$ ) is the probability space.

Ⅱ. Multiplication Law

0x00 official

(1) For events $A_1,A_2$ , $P(A) > 0$

$P(A_1\cap A_2) = P(A_1)P(A_2|A_1)$

(2) For events $A_1,A_2,A_3$ , $P(A_1\cap A_2)>0$ when,

$P(A_1\cap A_2\cap A_3) = P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)$

(3) For events $A_1...A_n$ , $P(A_1\cap ...\cap A_{n-1}) > 0$ when,

$P(A_1\cap A_2\cap ...\cap A_n)=P(A_1)P(A_2|A_1)P(A_3|A_1\cap A_2)...P(A_n|A_1\cap ...\cap A_{n-1})$

0x01 proof

（1） $P(A_2|A_1) = \frac{P(A_1\cap A_2)}{P(A_1)}$

（2） $P(A_3 | A_1 \cap A_2) = \frac{P(A_1\cap A_2\cap A_3)}{P(A_1\cap A_2)}$

Ⅲ. Independence of events

0x00 independent (independent)

For an event $A,B$ , $P(A) > 0$ when, if $P(B) = P(B|A)$ , we say that the event is independent $B$ of the event . $A$

0x01 independent and dependent

For events $A,B$ , if $P(A\cap B) = P(A)P(B)$ , we say that A and B are independent of each other.

If A and B are not independent of each other, we say that A and B are dependent on each other .

(1) If $A$ and $B$ are independent of each other, $A$ and $B^c$ are also independent of each other.

(2) If $P(A) = 0$ and $P(B) > 0$ , $A$ and $B$ are independent of each other.

(3) If $P(A) >0,P(B)>0$ and are mutually exclusive, then $A$ and are mutually dependent. $B$ $A$ $B$