Abstract
The aim of this paper is to obtain three types of expressions for calculating the probability of implementing palindromic digit combinations on a finite equally possible combination of zeros and ones. When calculating the probability of implementation of palindromic digit combinations, the classical definition of probability is applied. The main results of the paper are formulated in the form of three theorems. Moreover, the consequences of these theorems and typical examples of calculating the probability of implementing palindromic digit combinations in a data string of binary code are considered. All formulated theorems and their consequences are accompanied by proofs. The obtained numerical results of the paper can be used in the analysis of numerical computer data written as a binary code string in BIN format files. It should also be noted that the combinatorial expressions described in the article for calculating the number of palindromic combinations of digits in the binary number system can be used in number theory and in various branches of computer science. The development of these results from the point of view of obtaining an expression for calculating the number of palindromic combinations of digits in the binary number system contained in two-dimensional data arrays is also of immediate theoretical and practical interest. However, these results are not presented in this work, but they can be considered in subsequent publications.
Highlights
The number theory most often presents a palindrome as a symmetric natural number
By applying the expression to calculate the probability P(B) k / 2n, we find that the probability of random generation of a palindromic combination and the discharge n-i of the binary system within the framework of an possible combination of zeros and ones of an even discharge n is equal to P(В) 2(n i)/2 / 2n 2 (n i)/2
This paper considers obtaining three types of expressions for calculating the probability of generating palindromic combinations of the binary system digits within the framework of finite possible combinations of zeros and ones
Summary
The number theory most often presents a palindrome as a symmetric natural number. It is known that the term "palindrome" was introduced in the 17th century (from ancient Greek. πάλιν — "back, again" and ancient Greek. δρóμος — "running, movement") [1]. Expressions for calculating the probability of implementation of palindromes of the binary number system within the framework of a finite probable combination of zeros and ones were first obtained in the paper [15]. Let us note that the author does not know publications containing the results of calculating the probability of implementing palindromic combinations of digits of the binary number system on finite possible combinations of zeros and ones. The aim of this paper is to obtain three types of expressions for calculating the probability of implementing palindromic combinations of the binary number system on finite possible combinations of zeros and ones. Before the formulation of the results for calculating the probability of random generation of palindromes of the binary number system on possible combinations of zeros and ones, let us consider the following lemma. By analogy, according to Lemma 1, the number of all combinations of binary system digits for n=3 is equal to
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