{"id":234,"date":"2019-06-26T23:07:22","date_gmt":"2019-06-26T23:07:22","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=234"},"modified":"2019-11-18T23:57:07","modified_gmt":"2019-11-18T23:57:07","slug":"drunken-walker-and-fly","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/drunken-walker-and-fly\/","title":{"rendered":"Drunken Walker and Fly"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Imagine a drunken person wandering on the number line who starts at 0, and then moves left or right (+\/-1) with probability 1\/2. What is the probability that the walker will eventually return to her starting point Answer: probability 1.<\/p>\n\n\n\n

What about a random walk in the plane, moving on the integer lattice points, with probability 1\/4 in each of the coordinate directions? What’s the chance of return to the starting point? Answer: also probability 1.<\/p>\n\n\n\n

OK, now what about a drunken fly, with 6 directions to move, probability 1\/6? Surprisingly, it is probable that the fly will never return to its start. In fact it only has probability around 1\/3 of ever returning.<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Try to give a little insight by illustrating a random walk on the line for several steps.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
A probabilist would say that simple random walks on the line and plane are recurrent<\/em>, meaning that with probability one the walker would return to his starting point, and that simple random walks in dimensions 3 and higher are transient<\/em>, meaning there is a positive probability that he will never return! This is because there is so much “space” in dimensions 3 and higher.<\/p>\n\n\n\n

How to Cite this Page:<\/strong>\u00a0
Su, Francis E., et al. “Drunken Walker and Fly.”\u00a0Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by:<\/strong>
Lesley Ward<\/p>\n","protected":false},"excerpt":{"rendered":"

Imagine a drunken person wandering on the number line who starts at 0, and then moves left or right (+\/-1)...<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[3,45,146],"class_list":["post-234","page","type-page","status-publish","hentry","tag-easy","tag-probability","tag-random-walk"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=234"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/234\/revisions"}],"predecessor-version":[{"id":1412,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/234\/revisions\/1412"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=234"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}