# Well defined probability

Given the following probabilistic model

where $u$ are users of a search engine, $c$ are categories where the queries that $u$ search ($q$) and webs sites that they visit ($w$) are classified.

My question is if the following has necessarily has to hold for $p(c\mid q)$ to be a probability:

$$\displaystyle\sum_{i=1}^n\displaystyle\sum_{j=1}^m p(c_i\mid q_j) = 1.$$

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If $p(c\mid q)$ is a probability, then the following condition must be true: $$\displaystyle\sum_{i=1}^n\displaystyle\sum_{j=1}^m p(c_i\mid q_j) = 1.$$
As stated, that's not correct. For a fixed value of $q$, $p(c\mid q)$ is a probability on $c$. An informal definition of this is "The probability of $c$ occurring given that $q$ has already occurred." So, the following is true: $$\displaystyle\sum_{i=1}^n p(c_i\mid q_j) = 1.$$