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Yttrium
Can somebody explain me the sense of using the Baye's Law of probability and the total probability theorem? Thanks.
Suppose you have a tree of possible events like the one below: |dw:1385668006646:dw| Then, Baye's law says that the probability of one event on the far right (say \(\alpha\)), given that one of the preliminary events (say \(A\)) has occurred, is given by \[P(\alpha|A)=\frac{P(A|\alpha)P(\alpha)}{P(A)}=\frac{P(A|\alpha)P(\alpha)}{P(A|\alpha)+P(A|\beta)+P(A|\Gamma)}\] Basically, it says that the probability of some event \(A\) occurring, given the occurrence of another event \(\alpha\), is given by the ratio of (1) [the probability of \(A\) and \(\alpha\) occurring together] to (2) [the total probabilities of \(A\) occurring]. (1) The probability of two events occurring together is \(P(A\cap\alpha)\). Using the conditional probability definition, we get \(P(A|\alpha)=\dfrac{P(A\cap\alpha)}{P(\alpha)}\), i.e. \(P(A\cap\alpha)=P(A|\alpha)P(\alpha)\). (2) The total probability theorem is another way of saying that the denominators in the above equation are the same.