Statistics 简明教程

Statistics - Probability Additive Theorem

For Mutually Exclusive Events

概率加成定理指出，如果 A 和 B 是两个互斥事件，那么 A 或 B 的概率由下式给出

The additive theorem of probability states if A and B are two mutually exclusive events then the probability of either A or B is given by

该定理还可以扩展到三个互斥事件，如下所示

The theorem can he extended to three mutually exclusive events also as

Example

Problem Statement:

Problem Statement:

从 52 张牌中抽出一张，抽到国王或王后的概率是多少？

A card is drawn from a pack of 52, what is the probability that it is a king or a queen?

Solution:

Solution:

设事件 (A) = 抽出一张国王牌

Let Event (A) = Draw of a card of king

事件 (B) 抽出一张王后牌

Event (B) Draw of a card of queen

P（抽出的牌是国王或王后）= P（抽出的牌是国王）+ P（抽出的牌是王后）

P (card draw is king or queen) = P (card is king) + P (card is queen)

For Non-Mutually Exclusive Events

如果存在两种事件都发生的可能性，则加成定理写为：

In case there is a possibility of both events to occur then the additive theorem is written as:

Example

Problem Statement:

Problem Statement:

已知一个射手在 7 次射击中击中靶标 3 次；另一个射手在 5 次射击中击中靶标 2 次。当他们俩都尝试时，击中靶标的概率是多少？

A shooter is known to hit a target 3 out of 7 shots; whet another shooter is known to hit the target 2 out of 5 shots. Find the probability of the target being hit at all when both of them try.

Solution:

Solution:

第一个射手击中目标的概率 P(A) = ${\frac{3}{7}}$

Probability of first shooter hitting the target P (A) = ${\frac{3}{7}}$

第二个射击者击中目标的概率 P(B) = ${\frac{2}{5}}$

Probability of second shooter hitting the target P (B) = ${\frac{2}{5}}$

由于两个射击者都可能击中目标，因此事件 A 和 B 并非互斥。因此，适用的加法规则为

Event A and B are not mutually exclusive as both the shooters may hit target. Hence the additive rule applicable is