Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$P(U_{i+1}>U_i)=x_i$ for $i=1,\dots,n-1$ and $P(U_1>U_n)=x_n$. We call the
tuple $(x_1,\dots,x_n)$ \textit{nontransitive} if it is cyclic and in addition
satisfies $x_i>1/2$ for all $i$.
  Let $p_n$ (resp.~$p_n^*$) denote the probability that a randomly chosen
$n$-tuple $(x_1,\dots,x_n)\in[0,1]^n$ is cyclic (resp.~nontransitive). We
determine $p_3$ and $p_3^*$ exactly, while for $n\ge4$ we give upper and lower
bounds for $p_n$ that show that $p_n$ converges to $1$ as $n\to\infty$. We also
determine the distribution of the smallest, middle, and largest elements in a
cyclic triple.

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The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc