http://eprob.math.nsysu.edu.tw/ProbConcept/ConditionProb/index.html 在前面的機率介紹中，我們在一樣本空間上定義機率，有時得到一些資訊則根據所獲得的 資訊，要修訂樣本空間，因而機率空間可能也會改變，這就是所謂的條件機率 (conditional probability)。 在數學裡不會有這種情況。給定某個數是2，它就一直是2 ，在機率裡，某事件的機率是有可能因情況而變，這本來是不奇怪的， 但因大部分的人 受數學的薰陶較久，而數學裡通常是處理"不變"的問題，所以在學習機率時 ，看到機率 值居然會改變，便不易理解。 假設生男生女的機率各為1/2。則隨機抽取一個學生會是男 或女的機率也就大約是1/2。但若知此學生是高雄女中的學生， 則會是女生的機率就是1 了，因高雄女中沒有收男學生。由於獲得資訊，機率隨之而變，其實是合理的，否則就失 去收集資訊的目的。受機率的訓練，會使我們具有隨機的概念，對事物的研判，便能與時 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. 　設A、B為樣本空間中二事件，且P(B)>0。則在給定B發生之下，A之條件機 率，以P(A|B)表之，定義為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). 條件機率的定義中，B成為新的樣本空間 : P(B|B)=1。也就是原先的樣本空間 修正為B。所有事件發生之機率，都要先將其針對與B的關係作修正。例如，若與A 為B互斥事件，且P(B)>0，則因P(A∩B)=0，故P(A|B)=0，若P(A)亦為正，則此時亦有 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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