{"id":191,"date":"2022-01-19T05:45:58","date_gmt":"2022-01-19T05:45:58","guid":{"rendered":"https:\/\/qpr.ca\/blogs\/mathstuff\/?p=191"},"modified":"2022-01-19T05:45:58","modified_gmt":"2022-01-19T05:45:58","slug":"3-alan-coopers-answer-to-alice-and-bob-flip-a-biased-coin-best-mathn-math-out-of-math2n-1-math-win-if-the-probability-of-alice-winning-a-flip-is-mathp-math-what-is-her-chance-of-win","status":"publish","type":"post","link":"https:\/\/qpr.ca\/blogs\/mathstuff\/2022\/01\/19\/3-alan-coopers-answer-to-alice-and-bob-flip-a-biased-coin-best-mathn-math-out-of-math2n-1-math-win-if-the-probability-of-alice-winning-a-flip-is-mathp-math-what-is-her-chance-of-win\/","title":{"rendered":"(3) Alan Cooper&#8217;s answer to Alice and Bob flip a biased coin, best [math]n[\/math] out of [math]2n-1[\/math] win. If the probability of Alice winning a flip is [math]p[\/math], what is her chance of winning the series? &#8211; Quora"},"content":{"rendered":"<p>Since this is tagged with \u201cPuzzles and Trick Questions\u201d it may be that I am missing something. But my answer would be [math]\\Sigma_{m=0}^{n-1}p^{2n-1-m}(1-p)^m[\/math] .<\/p>\n<p>This follows the pattern of the best 2 out of 3 case where Alice has to win either two or three games &#8211; which happens in cases lww,wlw,wwl or www with probability [math]3p^2(1\u2013p)+p^3=3p^2\u20132p^3[\/math] (where the fact that the game may be stopped when she wins twice just corresponds to the fact that [math]pp(1-p)+ppp=p^2[\/math] , and the same answer is obtained by taking the complement of the cases where Bob wins either 2 or 3 games).<\/p>\n<p>Source: <em><a href=\"https:\/\/www.quora.com\/Alice-and-Bob-flip-a-biased-coin-best-n-out-of-2n-1-win-If-the-probability-of-Alice-winning-a-flip-is-p-what-is-her-chance-of-winning-the-series\/answer\/Alan-Cooper-5\">(3) Alan Cooper&#8217;s answer to Alice and Bob flip a biased coin, best [math]n[\/math] out of [math]2n-1[\/math] win. If the probability of Alice winning a flip is [math]p[\/math], what is her chance of winning the series? &#8211; Quora<\/a><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Since this is tagged with \u201cPuzzles and Trick Questions\u201d it may be that I am missing something. But my answer would be [math]\\Sigma_{m=0}^{n-1}p^{2n-1-m}(1-p)^m[\/math] . This follows the pattern of the best 2 out of 3 case where Alice has to win either two or three games &#8211; which happens in cases lww,wlw,wwl or www with&hellip;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-191","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/191","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/comments?post=191"}],"version-history":[{"count":1,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/191\/revisions"}],"predecessor-version":[{"id":192,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/191\/revisions\/192"}],"wp:attachment":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/media?parent=191"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/categories?post=191"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/tags?post=191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}