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I am a newbie in probability theory and following is my question:

Consider we have two binary random variables A and B. B is dependent on A. So we have two conditional probability tables P(A) and P(B|A) with the following parameters :

A P(A)

----------

F 0.3

T 0.7

A P(B=T|A)

------------------

F 0.4

T 0.6

Suppose that A=T is observed. So, now the probability of A being True is 1.0 instead of 0.3 and P(A=F) = 0.0 instead of 0.7. Observing A=T the probability of B=T is going to be 0.4, by just looking up the corresponding tuple in B's CPT. Consider a different scenario where we now observe only B=T. My first question is,

1) is P(B=T) going to be 1.0, since we have observed it, comparing to the first scenario of observing A=T? I know that if nothing is observed then P(B=T) is calculated as P(A=T)xP(B=T|A=T)+P(A=F)xP(B=T|A=F), which is the marginal probability of B=T.

My second question which follows from the first one is,

2) if P(B=T) = 1.0 when B=T has been observed, then while calculating the posterior probability of A=T given B=T i.e. P(A=T|B=T) why is it that we don't put P(B=T)=1.0 in the denominator of the following Baye's rule

P(A=T|B=T) = P(A=T) x P(B=T|A=T) / P(B=T)

Why do we use the marginal value of P(B=T) [when nothing is observed] computed by the expression

P(A=T) x P(B=T|A=T) + P(A=F) x P(B=T|A=F)?

Thanks in advance.

newbie

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# Question regarding probability of observation

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