Question Regarding Multi-label probability predictions
A question is a bit broaden as you do not specify if you do not know how to do it in theory or how to tackle it with a ML method.

Some of ML methods:

LogisticRegression in sklearn handles multiple class

lr = LogisticRegression()
lr.fit(X, y)
class_probabilities = lr.predict_proba(X)  # outputs the probabilities


You might also want to consider Support Vector Machines.

Theory: 

You can do a "one vs rest" when you train a single classifier per class taking the sample of all other classes as negative example. (see wiki article for that)