‎let $g$ be a finite group and $d_2(g)$ denotes the probability ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $g$‎. ‎we ‎will obtain lower and upper bounds for $d_2(g)$ in the case where ‎the sets $e_g(x)={yin g:[y,x,x]=1}$ are subgroups of $g$ for ‎all $xin g$‎. ‎also the given examples illustrate that all the ‎bounds are sharp.

According to the philosophical theory of inferentialism and its psychological counterpart, Hypothetical Inferential Theory (HIT), meaning an indicative conditional centrally involves strength inferential connection between antecedent consequent. This paper states, for first time, implications HIT probabilities conditionals. We report two experiments comparing these with those suppositional acco...