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