  This paper shows that one often can use only probabilistic estimates in many
cases for the analysis of the distribution of primes on the natural numbers and
arithmetic progressions, and also for study the distribution of prime $k$ -
tuples and prime values of the polynomials on the natural numbers. The author
generalizes Cramer's model to describe Hardy- Littlewood (concerning $k$ -
tuple) and Bateman-Horn conjectures. In the paper probabilistic models for
estimation of the accuracy of these conjectures were constructed. The author
proved performance using of the central limit theorem in the form of Lyapunov
for these probabilistic models. He found and proved the probabilistic estimates
of the accuracy for Hardy-Littlewood and Bateman-Horn conjectures and showed
the validity of these estimates in various cases.
