Perception of probability of being right The probability that people percept may be different from the real one due to a number of factors, including the form in which their are presented, their context and biases (due to misinformation or wishful thinking).

My question if there is any research showing the subjective probability judgement of being correct ($p_\mathrm{perceived}$) as a function of actual probability of correctness ($p_\mathrm{real}$) for situation were people estimate the chance that they are right in a test/trial (e.g. in a two answers forced choice task)?

Also, I am interested mathematical models with foundation in experimental data.
You'd probably want to take a look at signal detection theory. This is a set of tools that you use to analyse how well a person (or an animal, or a machine) is able to discern trials where there is a signal present (e.g. a picture is shown that has been shown before, a tumour is present in an x-ray image, a dot on a radar is an enemy aircraft) from trials where there is no-signal present (e.g. a picture is shown that hasn't been shown before, a tumour is not present in an x-ray image, a dot on a radar is not an enemy aircraft).

Whether a person perceives a signal or not is not always easy to discern. There will always be a certain interplay between the hit rate and the false alarm rate (in that you for example, if you always answered "signal", would have a perfect hit rate, although your false alarm rate would be terrible), and based on a number of criterias (time pressure, instructions, reward structure et cetera) one can manipulate the response patterns of a certain individual.

There has been attempts to create models where accuracy, speed and certainty all are taken into concideration; see for example Plescak and Busemeyer (2010).