# Intepretation of a probabilistic classifier's result

I understand that a probabilistic classifier predicts the probability distribution $$P(C|X)$$. However I do not understand what $$P(C|X)$$ means.

Is it "$$P(C=c|x)$$ the probability of belonging to class $$c$$ given the input $$x \in X$$"? If so then what is the sample sample?

I have done some researches and the term "calibration" comes up and it says that "$$P(C=c|x) = p$$ means that the $$p$$ is the probability of being correct for all inputs with the same output $$p$$". Is this the definition of the prediction or calibration is just a "metric" we are measuring our prediction?

Below is the population distribution of throwing two dices and to illustrate how to calculate the conditional probability.

In terms of using conditional probability to measure the accuracy of prediction/classification. In other words, it can be called Calibration - refers to the agreement between observed outcomes and predictions:

Assuming you are building two models to recognize the pictures of dogs, Your testing data is 5000 pictures of dogs and cats.

P(predict from model 1 = dog| testing = dog) = 0.99

P(predict from model 1 = dog | testing = cat) = 0.88

P(predict from model 2 = dog | testing = dog) = 0.89

P(predict from model 2 = dog | testing = cat) = 0.10

So, model 1 is overfitted and we prefer model 2. It means the P(predict|testing) can be used to measure the accuracy of the model.