1. 概率的性质

(1)对任何事件 $A, 0 \leq P (A) \leq 1$

(2) $P (Ω) = 1, P (\emptyset) = 0$

(3)设 $A$ 为任一随机事件,则 $P (\overset{ˉ}{A}) = 1 - P (A)$

(4) 设 $A \subset B$ ,则 $P (B - A) = P (B) - P (A)$

(5)设事件 $A_{1}, A_{2}, \dots, A_{n}$ 两两互斥,则

P (A_{1} + A_{2} + \dots + A_{n}) = P (A_{1}) + P (A_{2}) + \dots + P (A_{n})

(6)设 $A, B$ 为任意两个随机事件,则 $P (A \cup B) = P (A) + P (B) - P (A B)$

上式还能推广到多个事件的情况. 例如,设 $A_{1}, A_{2}, A_{3}$ 为任意三个事件,则有

$P (A_{1} \cup A_{2} \cup A_{3})$

= P (A_{1}) + P (A_{2}) + P (A_{3}) - P (A_{1} A_{2}) - P (A_{1} A_{3}) - P (A_{2} A_{3}) + P (A_{1} A_{2} A_{3})

一般,对于任意 $n$ 个事件 $A_{1}, A_{2}, \dots, A_{n}$

$P (A_{1} \cup A_{2} \cup \dots \cup A_{n})$

$= i = 1 \sum n P (A_{i}) - 1 \leq i \leq j \leq n \sum P (A_{i} A_{j}) + 1 \leq i \leq j \leq k \leq n \sum P (A_{i} A_{j} A_{k}) + \dots + (- 1)^{n - 1} P (A_{1} A_{2} \dots A_{n}) .$

Youliang Zhong

Backlinks

Graph View

1. 概率的性质

1. 概率的性质