  We prove precise deviations results in the sense of Cram\'er and Petrov for
the upper tail of the distribution of the maximal value for a special class of
determinantal point processes that play an important role in random matrix
theory. Here we cover all three regimes of moderate, large and superlarge
deviations for which we determine the leading order description of the tail
probabilities. As a corollary of our results we identify the region within the
regime of moderate deviations for which the limiting Tracy-Widom law still
predicts the correct leading order behavior. Our proofs use that the
determinantal point process is given by the Christoffel-Darboux kernel for an
associated family of orthogonal polynomials. The necessary asymptotic
information on this kernel has mostly been obtained in [Kriecherbauer T.,
Schubert K., Sch\"uler K., Venker M., Markov Process. Related Fields 21 (2015),
639-694]. In the superlarge regime these results of do not suffice and we put
stronger assumptions on the point processes. The results of the present paper
and the relevant parts of [Kriecherbauer T., Schubert K., Sch\"uler K., Venker
M., Markov Process. Related Fields 21 (2015), 639-694] have been proved in the
dissertation [Sch\"uler K., Ph.D. Thesis, Universit\"at Bayreuth, 2015].
