{"id":4857,"date":"2026-05-09T09:53:56","date_gmt":"2026-05-09T13:53:56","guid":{"rendered":"https:\/\/josmfs.net\/wordpress\/?p=4857"},"modified":"2026-05-09T09:53:56","modified_gmt":"2026-05-09T13:53:56","slug":"knock-the-bottle-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2026\/05\/09\/knock-the-bottle-problem\/","title":{"rendered":"Knock the Bottle Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-4854\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/05\/Knock-the-Bottle-Fig2.jpg\" alt=\"\" width=\"201\" height=\"170\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/05\/Knock-the-Bottle-Fig2.jpg 403w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/05\/Knock-the-Bottle-Fig2-300x254.jpg 300w\" sizes=\"auto, (max-width: 201px) 100vw, 201px\" \/>This is a <a href=\"https:\/\/medium.com\/math-games\/are-you-smarter-than-a-6-year-old-3e73146caded\">problem<\/a> from Barry Leung\u2019s Math Games.<\/p>\n<p>\u201cLarry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge.\u00a0 Larry throws first. The winner is the first person to knock the bottle off the ledge.\u00a0 At each turn the probability that a player knocks the bottle off the ledge is \u00bd, independently of what has happened before.\u00a0 What is the probability that Larry wins the game?\u201d<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/05\/Knock-the-Bottle-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2026\/05\/Knock-the-Bottle-260406.pdf\">Knock the Bottle Problem<\/a> for a solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a problem from Barry Leung\u2019s Math Games. \u201cLarry and Julius are playing a game, taking turns throwing a ball at a bottle sitting on a ledge.\u00a0 Larry throws first. The winner is the first person to knock the bottle off the ledge.\u00a0 At each turn the probability that a player knocks the bottle [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[274,138,230],"class_list":["post-4857","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-bl-puzzles","tag-probability","tag-sequences-and-series"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4857","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=4857"}],"version-history":[{"count":1,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4857\/revisions"}],"predecessor-version":[{"id":4858,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/4857\/revisions\/4858"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=4857"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=4857"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=4857"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}