tag:blogger.com,1999:blog-7905515.post8773632229925573191..comments2024-08-17T10:51:33.672-05:00Comments on Falkenblog: The Birthday ProblemEric Falkensteinhttp://www.blogger.com/profile/07243687157322033496noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-7905515.post-89144974134457835762008-07-22T11:45:00.000-05:002008-07-22T11:45:00.000-05:00Though the practical significance may be zero, the...Though the practical significance may be zero, there is some confusion on this, at least in my mind:<BR/>Using the combination approach assumes the pairs are independent, which they are not. This is usually a second order error. for example, Using the simple # of pairs approach, the probability no two people share a birthday is (364/365)^(n*(n-1)/2)<BR/>or <BR/><BR/>n(n-1)/2=n!/(2!(n-2)!)=C(n choose 2)<BR/><BR/>For 23, this is 0.4995<BR/>so prob against, 0.5005<BR/><BR/>The 100% correct approach uses permutations, where the answer is P(365,23)/365^23 or 0.493, for a prob against of 0.507Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-46271674295790833242008-07-22T11:27:00.000-05:002008-07-22T11:27:00.000-05:00I get 1-.462=.538 as the chance that someone has h...I get 1-.462=.538 as the chance that someone has he same birthday. <BR/><BR/>But a more relevant questions is ... shouldn't I be doing something else rather than thinking about this?Pete Shttps://www.blogger.com/profile/01722037274235500118noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-64534705896839042692008-07-22T11:16:00.000-05:002008-07-22T11:16:00.000-05:00Maybe I missed something, but shouldn't it be:364!...Maybe I missed something, but shouldn't it be:<BR/>364!/{(365**23)*(341!)}Pete Shttps://www.blogger.com/profile/01722037274235500118noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-76386375145470134892008-07-21T20:16:00.000-05:002008-07-21T20:16:00.000-05:00oops. I put n*p, not p^n. txoops. I put n*p, not p^n. txEric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-76206214302259292032008-07-21T20:03:00.000-05:002008-07-21T20:03:00.000-05:00Eric, see my math below. why is it wrong?? ((23^2...Eric, <BR/>see my math below. why is it wrong?? <BR/><BR/>((23^2) / 2) * (364 / 365) = 263.775342<BR/><BR/>thanks, DanAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-7905515.post-44389285989820590302008-07-21T17:35:00.000-05:002008-07-21T17:35:00.000-05:00It's an approximation. So, so 23 people not shari...It's an approximation. So, so 23 people not sharing a birthday is 23^2/2*(364/365)=48.4%, and so the probability at least 2 people share a birthday is 1-48.4%, or 51.6%. Because of adjustments outside the scope of this post, the actual number is 50.7%--close enough.Eric Falkensteinhttps://www.blogger.com/profile/07243687157322033496noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-89825575320727341652008-07-21T17:21:00.000-05:002008-07-21T17:21:00.000-05:00Arguing about this stuff seems especially silly to...Arguing about this stuff seems especially silly to me in a system that routinely relies on eye-witness testimony as the gold standard of proof. We debate whether the DNA false positive rate is one-in-a-billion or merely one-in-a-million; meanwhile we we send people to prison for years based on witness testimony that is probably more like one-in-twenty, if you're lucky.edhttps://www.blogger.com/profile/01150091053740909530noreply@blogger.comtag:blogger.com,1999:blog-7905515.post-31756201613040408212008-07-21T17:17:00.000-05:002008-07-21T17:17:00.000-05:00(N^2)/2*(364/365)Are you sure you have your math r...(N^2)/2*(364/365)<BR/><BR/>Are you sure you have your math right here?Johnhttps://www.blogger.com/profile/01457388998903348000noreply@blogger.com