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If I have given a complete table for the joint probability $$P(A,B,C,D,E)$$ how can I compute an arbitrary conditional probability out of it, for instance: $$P(A|B)$$

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2 Answers 2

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$$\mathbb{P}(A=a \vert B=b) = \frac{\mathbb{P}(A=a, B=b)}{\mathbb{P}(B=b)} = \frac{\displaystyle \sum_{c,d,e} \mathbb{P}(A=a, B=b, C=c, D=d, E=e)}{\displaystyle \sum_{a,c,d,e} \mathbb{P}(A=a, B=b, C=c, D=d, E=e)}$$

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The short answer for your example is that you can compute $P(A,B)$ and $P(B)$ from the table (you have to sum out all the other variables for fixed A and B). Using these values you can compute $P(A|B)$.

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  • $\begingroup$ For instance for $A = a$ and $B = b$ it would be: $$P(A = a, B = b) = \sum_{C,D,E} P(A = a, B = b, C, D ,E)$$? $\endgroup$
    – Mahoni
    May 21, 2012 at 18:49
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    $\begingroup$ @Mahoni Exactly. $\endgroup$
    – svenkatr
    May 21, 2012 at 18:50

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