{"pk":64982,"title":"The probability that a random triple of dice is transitive","subtitle":null,"abstract":"An \\(n\\)-sided die is an \\(n\\)-tuple of positive integers. We say that a die \\((a_1,\\dots,a_n)\\) beats a die \\((b_1,\\dots,b_n)\\) if the number of pairs \\((i,j)\\) such that \\(a_i›b_j\\) is greater than the number of pairs \\((i,j)\\) such that \\(a_i‹b_j\\). We show that for a natural model of random \\(n\\)-sided dice, if \\(A, B\\) and \\(C\\) are three random dice then the probability that \\(A\\) beats \\(C\\) given that \\(A\\) beats \\(B\\) and \\(B\\) beats \\(C\\) is approximately 1/2. In other words, the information that \\(A\\) beats \\(B\\) and \\(B\\) beats \\(C\\) has almost no effect on the probability that \\(A\\) beats \\(C\\). This proves a statement that was conjectured by Conrey, Gabbard, Grant, Liu and Morrison for a different model.\n \nMathematics Subject Classifications: 60C05\n \nKeywords: Intransitive dice, central limit theorems","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Intransitive dice"},{"word":"central limit theorems"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/62c6g1zm","frozenauthors":[{"first_name":"D. H. J.","middle_name":"","last_name":"Polymath","name_suffix":"","institution":"","department":""}],"date_submitted":"2025-07-15T15:03:10Z","date_accepted":"2025-07-15T15:03:10Z","date_published":"2025-07-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64982/galley/49792/download/"}]}