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I have: P(c) = 0.98 and P(v) = 0.96. Also I have this conditional probability table:

C   V   P(p|c,v)
-----------------
T   V   0.99
T   F   0.01
F   V   0
F   F   0

(T = True, F = False)

My question is: how to calculate P(p|c) ?

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To approach this problem, consider the total law of probability. We can express $Pr(p|c=c')$ as

$$ Pr(p|c=c') = \sum_n Pr(p|c=c' \cap V_n) Pr(V_n | c=c') $$

So then by inspection,

$$ Pr(p|c=False) = 0 $$

Now from here we can make various assumptions about the distribution of $C$ and $V$, such as independence, but to me there isn't enough information to solve this problem.


If we assume independence between $C$ and $V$, we can try to solve the problem, by saying:

$$ Pr(p|c=True) = \sum_n Pr(p|c=True \cap V_n) Pr(Vn) = 0.99*0.96 + 0.01*0.04 $$

what is interesting about this, is that if we assume independence, then the information that $Pr(C) = 0.98$ is in fact not even required, since the conditional probabilities are provided.

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