We start with the famous Heisenberg uncertainty principle to give the idea of the probability in quantum mechanics. The Heisenberg uncertainty principle states by precise inequalities that the product of uncertainties of two physical quantities, such as momentum and position (operators), must be greater than certain (strictly positive) constant, which means that if we know one of the quantities more precisely, then we know the other one less precisely. Therefore, in quantum mechanics, predictions should be probabilistic, not deterministic, and then position and momentum should be considered as random variables to measure their probabilities.
In mathematical framework, the noncommutative probability is another name of quantum probability, and a quantum probability space consists of an -algebra of operators on a Hilbert space and a state (normalized positive linear functional) on the operator algebra. We study the basic notions in quantum probability theory comparing with the basic notions in classical (commutative) probability theory, and we also study the fundamental theory of quantum stochastic calculus motivated by the classical stochastic calculus.
Finally, we discuss several applications with future prospects of classical and quantum probability theory.