Statistics 简明教程

Statistics - Probability Multiplicative Theorem

For Independent Events

定理指出，两个独立事件同时发生的概率等于其各自概率的乘积。

The theorem states that the probability of the simultaneous occurrence of two events that are independent is given by the product of their individual probabilities.

该定理也可以扩展到三个或更多独立事件，如下所示

The theorem can he extended to three or more independent events also as

Example

Problem Statement:

一所大学必须聘用一位讲师，该讲师必须拥有学士学位、MBA 和博士学位，其概率分别为 ${\frac{1}{20}}$、${\frac{1}{25}}$ 和 ${\frac{1}{40}}$。找出大学聘请这样一位人员的概率。

A college has to appoint a lecturer who must be B.Com., MBA, and Ph. D, the probability of which is ${\frac{1}{20}}$, ${\frac{1}{25}}$, and ${\frac{1}{40}}$ respectively. Find the probability of getting such a person to be appointed by the college.

Solution:

某人成为学士的概率 P(A) = ${\frac{1}{20}}$

Probability of a person being a B.Com.P(A) =${\frac{1}{20}}$

某人成为 MBA 的概率 P(B) = ${\frac{1}{25}}$

Probability of a person being a MBA P(B) = ${\frac{1}{25}}$

一个人拥有博士学位的概率 P© =${\frac{1}{40}}$

Probability of a person being a Ph.D P© =${\frac{1}{40}}$

使用独立事件的乘法定理

Using multiplicative theorem for independent events

For Dependent Events (Conditional Probability)

如前所定义，相关事件指的是一个事件的发生或不发生会影响下一个事件的结果。对于此类事件，前面陈述的乘法定理不适用。与这种事件相关的概率称为条件概率，由下式给出：

As defined earlier, dependent events are those were the occurrences or nonoccurrence of one event effects the outcome of next event. For such events the earlier stated multiplicative theorem is not applicable. The probability associated with such events is called as conditional probability and is given by

P(A/B) = ${\frac{P(AB)}{P(B)}}$ 或 ${\frac{P(A \cap B)}{P(B)}}$

P(A/B) = ${\frac{P(AB)}{P(B)}}$ or ${\frac{P(A \cap B)}{P(B)}}$

P(A/B) 被读作在事件 B 已经发生时事件 A 发生的概率。

Read P(A/B) as the probability of occurrence of event A when event B has already occurred.

类似地，给定 A 的 B 的条件概率为：

Similarly the conditional probability of B given A is

P(B/A) = ${\frac{P(AB)}{P(A)}}$ 或 ${\frac{P(A \cap B)}{P(A)}}$

P(B/A) = ${\frac{P(AB)}{P(A)}}$ or ${\frac{P(A \cap B)}{P(A)}}$

Example

Problem Statement:

一枚硬币被抛掷 2 次。抛掷结果为一次正面和一次反面。第一次抛掷为反面的概率是多少？

A coin is tossed 2 times. The toss resulted in one head and one tail. What is the probability that the first throw resulted in a tail?

Solution:

抛掷两次硬币的样本空间给定为 S = {HH, HT, TH, TT}

The sample space of a coin tossed two times is given as S = {HH, HT, TH, TT}

设事件 A 为第一次抛掷结果为反面。

Let Event A be the first throw resulting in a tail.

事件 B 为发生一次反面和一次正面。

Event B be that one tail and one head occurred.