  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 our former
works [K. Tadaki, arXiv:1804.10174], based on the toolkit of algorithmic
randomness, we presented a refinement of the Born rule, called the principle of
typicality, for specifying the property of results of measurements in an
operational way. In this paper, we make an application of our framework to the
argument of local realism versus quantum mechanics for refining it, in order to
demonstrate how properly our framework works in practical problems in quantum
mechanics.
