70% certain with a 70% success rate We have some classifiers that classify text documents. Each classifier reports how likely their classification is correct based on what they know of the document. Classifiers do not know how accurate they themselves are. 
If a classifier reports that it is 70% likely that document belongs to class A and that classifier is correct in its classifications 70% of the time, what is the likelihood that the document actually is a member of class A. Surely not 70% of 70%, so what would the actual calculation be?
 A: You simply don't have enough information. 
According to the usual definitions, the $70\%$ accuracy rate of the classifier is not the average of the probabilities assigned to the correct class. It is the frequency with which the classifier predicts that the correct class has the highest probability. This can mean that the correct category is assigned a probability of $10\%$ while several other categories are assessed at $8\%$. You can mix your classifier with a clueless classifier, which assigns every category the same probability regardless of the inputs, and the mixture will have the same accuracy rate as the original, but it will make different probability estimates. 
A: You're looking for the precision. In the context of medical diagnosis, it is called the positive predictive value. 
As @image_doctor already commented, knowing the prior probabilities of your classes is crucial for calculating a meaningful precision.
The classifier saying it is 70% sure that the document belong to class A is rather irrelevant in this context: the precision is a proportion of "hard" classifications, i.e. the classifier must say it is A or it isn't. The 70% posterior class membership of the classifier must therefore  be converted into such a hard assignment first. You need a rule for that such as a threshold (working point). You can also calculate precision as function of the threshold, similar to what is done for constructing the ROC. 
Side note: There are possibilities to calculate the precision directly from the continuous output (you can nose around here, and I have a manuscript waiting to be put into ArXiv, but I need it to be accepted by the journal first). However, if just the "normal" precision is asked for, that would be rather a sledgehammer to crack the nut.
