{"id":3671,"date":"2023-11-23T08:18:49","date_gmt":"2023-11-23T13:18:49","guid":{"rendered":"https:\/\/sites.williams.edu\/Morgan\/?page_id=3671"},"modified":"2023-11-27T15:56:53","modified_gmt":"2023-11-27T20:56:53","slug":"how-to-play-fair-with-loaded-dice","status":"publish","type":"page","link":"https:\/\/sites.williams.edu\/Morgan\/math-chat-archives\/how-to-play-fair-with-loaded-dice\/","title":{"rendered":"How to Play Fair With Loaded Dice"},"content":{"rendered":"<p>September 7, 2000<\/p>\n<p>&nbsp;<\/p>\n<p><b>Questionable Mathematics.<\/b>\u00a0<i>The Christian Science Monitor<\/i>\u00a0of July 18 reported that &#8220;More than 650 million&#8221; Americans took to the skies last year. That&#8217;s more than twice the entire population! They probably meant 650 million passenger flights, counting some people many times.<\/p>\n<p>Math Chat invites readers to submit further examples of questionable mathematics.<\/p>\n<p><b>Old Challenge<\/b>\u00a0(Al Zimmermann). For some reason you need a standard fair die (with the numbers from 1 to 6 on the six faces). All you have is an unfair &#8220;loaded&#8221; die. How can you use the loaded die to fairly choose random numbers from 1 to 6?<\/p>\n<p><b>Answer\u00a0<\/b>(Jos\ufffd Ca\ufffdizo and Arthur Pasternak). Roll the die three times. If you don&#8217;t get three different numbers, start over. When you get three different numbers, the highest roll H, the middle roll M, and the lowest roll L can occur in six, equally likely patterns: HML, HLM, MHL, MLH, LHM, and LMH, which can be associated with the numbers from 1 to 6.<\/p>\n<p><b>New Challenge.<\/b>\u00a0In the September ACBL Bridge Bulletin, Noreen Wurdemann of the Bahamas reports that the two bridge hands below were dealt out at her club. She says that she is &#8220;100% certain there was no hanky-panky.&#8221; What is the probability of this happening among say 28 boards? somewhere in the world in a year? in a century? How do you explain this report?<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/sites.williams.edu\/Morgan\/files\/2023\/11\/BridgeHands.gif\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter  wp-image-3779\" src=\"https:\/\/sites.williams.edu\/Morgan\/files\/2023\/11\/BridgeHands-300x121.gif\" alt=\"\" width=\"434\" height=\"175\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<hr \/>\n<p>Send answers, comments, and new questions by email to\u00a0<a href=\"mailto:Frank.Morgan@williams.edu\">Frank.Morgan@williams.edu,<\/a>\u00a0to be eligible for<i>\u00a0Flatland\u00a0<\/i>and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan&#8217;s homepage is at\u00a0<a href=\"http:\/\/www.williams.edu\/Mathematics\/fmorgan\">www.williams.edu\/Mathematics\/fmorgan.<\/a><\/p>\n<p><a href=\"http:\/\/www.maa.org\/books\/mch.html\">THE MATH CHAT BOOK,<\/a>\u00a0including a $1000 Math Chat Book\u00a0<a href=\"http:\/\/www.maa.org\/books\/quest.html\">QUEST,\u00a0<\/a>questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).<\/p>\n<p>Copyright 2000, Frank Morgan.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>September 7, 2000 &nbsp; Questionable Mathematics.\u00a0The Christian Science Monitor\u00a0of July 18 reported that &#8220;More than 650 million&#8221; Americans took to the skies last year. That&#8217;s more than twice the entire population! They probably meant 650 million passenger flights, counting some people many times. Math Chat invites readers to submit further examples of questionable mathematics. Old [&hellip;]<\/p>\n","protected":false},"author":2965,"featured_media":0,"parent":3459,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-3671","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3671","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/2965"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=3671"}],"version-history":[{"count":3,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3671\/revisions"}],"predecessor-version":[{"id":3781,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3671\/revisions\/3781"}],"up":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3459"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=3671"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}