OICC Press Mathematical Sciences 2251-7456 13 1 (March 2019) 2019 01 02 Certain numerical results in non-associative structures 10.1007/s40096-018-0274-0 EN Behnam Azizi Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IR Hossein Doostie Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IR Journal Article 2019 01 02 Abstract The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document} , the n th-commutativity degree of a finite algebraic structure S , denoted by Pn(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_n(S)$$\end{document} , is the probability that for chosen randomly two elements x and y of S , the relator xny=yxn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^ny=yx^n$$\end{document} holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the n th-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for n th-commutativity degree of these loops, we will obtain best upper bounds for this probability.