On cyclic and nontransitive probabilities

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ such that $P(U_{i+1}>U_i)=x_i$ $i=1,\dots,n-1$ and $P(U_1>U_n)=x_n$. We the tuple $(x_1,\dots,x_n)$ \textit{nontransitive} it is cyclic in addition satisfies $x_i>1/2$ all $i$. Let $p_n$ (resp.~$p_n^*$) denote probability a randomly chosen $(x_1,\dots,x_n)\in[0,1]^n$ (resp.~nontransitive). determine $p_3$ $p_3^*$ exactly, while $n\ge4$ give upper lower bounds show converges to $1$ as $n\to\infty$. also distribution of smallest, middle, largest elements triple.