## Yttrium 2 years ago Can somebody explain me the sense of using the Baye's Law of probability and the total probability theorem? Thanks.

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1. SithsAndGiggles

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.