Can somebody explain me the sense of using the Baye's Law of probability and the total probability theorem? Thanks.

Question

anonymous · Answer

Suppose you have a tree of possible events like the one below:
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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.