{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T21:16:13Z","timestamp":1792790173029,"version":"4.5.0"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2017,4,7]],"date-time":"2017-04-07T00:00:00Z","timestamp":1491523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11461043,11361042"],"award-info":[{"award-number":["11461043,11361042"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the Provincial Natural Science Foundation of Jiangxi, China","award":["20142BAB201005"],"award-info":[{"award-number":["20142BAB201005"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Intuitionistic fuzzy probabilities are an extension of the concept of probabilities with application in several practical problem solving tasks. The former are probabilities represented through intuitionistic fuzzy numbers, to indicate the uncertainty of the membership and nonmembership degrees in the value assigned to probabilities. Moreover, a dual hesitant fuzzy set (DHFS) is an extension of an intuitionistic fuzzy set, and its membership degrees and nonmembership degrees are represented by two sets of possible values; this new theory of fuzzy sets is known today as dual hesitant fuzzy set theory. This work will extend the notion of dual hesitant fuzzy probabilities by representing probabilities through the dual hesitant fuzzy numbers, in the sense of Zhu et al., instead of intuitionistic fuzzy numbers. We also give the concept of dual hesitant fuzzy probability, based on which we provide some main results including the properties of dual hesitant fuzzy probability, dual hesitant fuzzy conditional probability, and dual hesitant fuzzy total probability.<\/jats:p>","DOI":"10.3390\/sym9040052","type":"journal-article","created":{"date-parts":[[2017,4,7]],"date-time":"2017-04-07T10:49:53Z","timestamp":1491562193000},"page":"52","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Dual Hesitant Fuzzy Probability"],"prefix":"10.3390","volume":"9","author":[{"given":"Jianjian","family":"Chen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Nanchang University, Nanchang 330031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xianjiu","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Nanchang University, Nanchang 330031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhu, B., Xu, Z., and Xia, M. (2012). Dual hesitant fuzzy sets. J. Appl.Math., 2012.","DOI":"10.1155\/2012\/879629"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","article-title":"Hesitant fuzzy information aggregation in decision making","volume":"52","author":"Xia","year":"2011","journal-title":"Int. J. Approx. Reason."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1016\/j.apm.2013.07.010","article-title":"Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making","volume":"38","author":"Ye","year":"2014","journal-title":"Appl. Math. Model."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.3233\/IFS-130845","article-title":"Some results for dual hesitant fuzzy sets","volume":"26","author":"Zhu","year":"2014","journal-title":"J. Intell. 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