http://eprob.math.nsysu.edu.tw/ProbConcept/ConditionProb/index.html 在前面的机率介绍中，我们在一样本空间上定义机率，有时得到一些资讯则根据所获得的 资讯，要修订样本空间，因而机率空间可能也会改变，这就是所谓的条件机率 (conditional probability)。 在数学里不会有这种情况。给定某个数是2，它就一直是2 ，在机率里，某事件的机率是有可能因情况而变，这本来是不奇怪的， 但因大部分的人 受数学的薰陶较久，而数学里通常是处理"不变"的问题，所以在学习机率时 ，看到机率 值居然会改变，便不易理解。 假设生男生女的机率各为1/2。则随机抽取一个学生会是男 或女的机率也就大约是1/2。但若知此学生是高雄女中的学生， 则会是女生的机率就是1 了，因高雄女中没有收男学生。由於获得资讯，机率随之而变，其实是合理的，否则就失 去收集资讯的目的。受机率的训练，会使我们具有随机的概念，对事物的研判，便能与时 推移。 In the previous introduction to probability, we define probabilty in a sample space. Sometimes, the sample space should be updated based on our newly obtained information, and therefore the probability space may also change. This is the so called conditional probability. There is not such thing in mathematics: if a given number is 2, it is always 2. In probability, the probability of an event may change in different circumstances. This is really not strange, but most people have long been influenced by an assumption that mathematics ususally involve "invariable" problems; thus, when studying mathematics, they see with surprise that probabilistic values would vary but cannot easily understand it. Suppose the probabilities of giving birth to a boy or a girl are both 1/2. Then a ramdomly chosen student would be a male or a female with about 1/2 probability. But if this student is known to be a student in Kaohsiung Girl's Senior High School, then the probability to be a female is just 1 because Kaohsiung Girl's Senior High School does not admit male students. Probabilities vary with the obtained information; this is infact very reasonable, or the purposes of collecting information are lost. Training in probability theory leads us to the concept of randomness, and we can go with the times in our judgment of things. 　设A、B为样本空间中二事件，且P(B)>0。则在给定B发生之下，A之条件机 率，以P(A|B)表之，定义为P(A|B) = P(A∩B) / P(B)。 Suppose A and B are two event in the sample space, and P(B) > 0. Given B happens, the conditional probability of A, denoted by P(A|B), is defined by P(A|B) = P(A∩B) / P(B). 条件机率的定义中，B成为新的样本空间 : P(B|B)=1。也就是原先的样本空间 修正为B。所有事件发生之机率，都要先将其针对与B的关系作修正。例如，若与A 为B互斥事件，且P(B)>0，则因P(A∩B)=0，故P(A|B)=0，若P(A)亦为正，则此时亦有 P(B|A)=0 In the definition of conditional probability, B becomes the new sample space: P(B|B) = 1. In other words, the original sample space is updated to B, and the probabilities of all events should also be updated according to their relationships with B. For example, if A and B are mutually exclusive and P(B) > 0, then P(A|B) = 0 because P(A∩B)=0. If P(A) is positive, then P(B|A) = 0, too. --

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