Why is quantum mechanics based on probability theory? What makes us formulate quantum mechanics based on probability theory?
Isn't the real quantum world based on unknown laws to us?
Is it possible that results of an experiment will be measurable in another way but not expected value?
 A: I'll have a go to show that the concept of probability is a mathematical tool for formulating a  theory of the mechanics that governs  the microcosm, which ended into Quantum mechanics as we know it.
To start with, what is probability theory in mathematics ?

Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena.1 The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted.

In contrast, Quantum Mechanics is a theory with dynamical solutions of specific differential equations with imposed physical boundary conditions. There is nothing random about these solutions. Thus QM is not based on probability theory as the events  are not random and are not from the distributions appearing in the studies of probability theory. 
It uses the concept of probability: the outcomes of particle experiments  repeated many times are predicted by the  QM solutions, an individual experiment having a calculable  probability of appearing taken from those solutions.
Quantum Mechanics became necessary because there were experiments where classical mechanics was not able to predict or explain them. Particles, entities with specific (x,y,z,t) and (p_x,p_y,p_z,E), exhibited also a wave nature, but not of a classical wave i.e.  of a medium reacting to energy passing. A new type of wave that appears in the probability space .
An example is the double slit experiment with individual electrons measured over time.

A probability distribution can be defined by these data, that has nothing to do with the probability theory's distribution and everything to do with the QM mathematical solution giving a wave function, whose square  gives the probability of an electron appearing at a given space point. 
A: First question: the origin of quantum mechanics can in certain sense be identified from the intuition of De Broglie that every particle could be described through a wave. 
Starting from this point, Schroedinger built the famous equation, to which several interpretation can be given.
Schroedinger himself at first thought that the equation described the density charge over an infinite volume in the space. 
Born instead gave the intepretation that now is generally accepted , that the equation described the probability distribution of the particle in the space. Please notice that these are postulates and they can be confirmed only empirically, with a lot of experimental work.
Second question: that is what Einstein thought about quantum theory. So far, all the experiments confirmed the stochastic nature of the quantum physics, so i would say no, it is not.
Third question: if you mean measured in a deterministic way, violating the Heisenberg uncertainty principle, I would say no, is not possible, for what we know so far.
