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I have got the following Network and porbabilites enter image description here

And I need to Calculate the probability $P(C=False)$

I triend using the formula for joint porability distributions $P(X_1,…,X_n)= \prod_{i=1}^{n}P(X_i∣ parents (X_i))$
,and got $P(C=False)=P(C=False|A,B)$.

If I am not wrong, you can infer that $P(C=False|A,B) = 0.75$ from the table of probabilites , but I would like to know what is the general way to solving a problem like this in case not all the probabilities for each case are the same.

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1 Answer 1

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This is the way to calculate:

$$\begin{align} P(C=False)&=\sum_{a,b} P(C=False,A=a,B=b)\\ &=\sum_{a,b}P(C=False|A=a,B=b)P(A=a, B=b)\\ &=\sum_{a,b}P(C=False|A=a,B=b)P(A=a)P(B=b) \end{align}$$

Here, $a, b$ can be $True, False$. Values in the summation can be substituted from the table.

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