  The notion of probability plays a crucial role in quantum mechanics. It
appears in quantum mechanics as the Born rule. In modern mathematics which
describes quantum mechanics, however, probability theory means nothing other
than measure theory, and therefore any operational characterization of the
notion of probability is still missing in quantum mechanics. In this paper,
based on the toolkit of algorithmic randomness, we present a refinement of the
Born rule, as an alternative rule to it, for specifying the property of the
results of quantum measurements in an operational way. Algorithmic randomness
is a field of mathematics which enables us to consider the randomness of an
individual infinite sequence. We then present an operational refinement of the
Born rule for mixed states, as an alternative rule to it, based on algorithmic
randomness. In particular, we give a precise definition for the notion of mixed
state. We then show that all of the refined rules of the Born rule for both
pure states and mixed states can be derived from a single postulate, called the
principle of typicality, in a unified manner. We do this from the point of view
of the many-worlds interpretation of quantum mechanics. Finally, we make an
application of our framework to the BB84 quantum key distribution protocol in
order to demonstrate how properly our framework works in practical problems in
quantum mechanics, based on the principle of typicality.
