It gets even more interesting when you ask about which of 2 triples of head/tail sequences appears first in an infinite sequence of heads and tails. Martin Gardiner wrote about this in the early 1970's Martin Gardner, "Mathematical Games: The Paradox of the Nontransitive Dice and the Elusive Principle of Indifference." Scientific American 223, 110-114, Dec. 1970 (and perhaps again in 1974). His book, "The Colossal Book of Mathematics: classic puzzles, paradoxes, and problems" has that stuff reprinted and updated.
Bill Dunlap TIBCO Software Inc - Spotfire Division wdunlap tibco.com > -----Original Message----- > From: r-help-boun...@r-project.org > [mailto:r-help-boun...@r-project.org] On Behalf Of Erik Iverson > Sent: Monday, August 31, 2009 1:35 PM > To: Erik Iverson; r-help@r-project.org > Subject: Re: [R] Offtopic, HT vs. HH in coin flips > > Part of my issue was that I was not answering my original > question. "What is more likely to show up first, HT or HH?" > The answer to that turns out to be "neither", or "identical chances". > > ht <- replicate(2500, > paste(sample(c("H", "T"), 100, replace = TRUE), > collapse = "")) > > hts <- regexpr("HT", ht) + 1 > hhs <- regexpr("HH", ht) + 1 > > ## which is first? > table(hts < hhs) # about 50/50 > > summary(hts) #mean of 4 > summary(hhs) #mean of 6 > > So, "What is more likely to show up first, HH or HT?" is of > course a different question than "Are the expected values of > the positions for the first HT or HH the same?" I suppose > that's where confusion set in. It seems that if HH appears > later in the string on average (i.e., after 6 tosses instead > of 4), that the probability of it being first would be lower > than HT, which is obviously wrong! > > A quick graphic that helps show this (you must run the above > code first): > > library(lattice) > > ht.df <- data.frame(count = c(hts, hhs), > type = gl(2, 1250, labels = c("HT", "HH"))) > > barchart(prop.table(xtabs(~ count + type, data = ht.df)), > stack = FALSE, horizontal = FALSE, > box.ratio = .8, auto.key = TRUE) > > Thanks to all those who replied, and also someone sent me the > following link off list, it also clears up the confusion: > > http://www.mit.edu/~emin/writings/coinGame.html > > Best, > Erik > > -----Original Message----- > From: r-help-boun...@r-project.org > [mailto:r-help-boun...@r-project.org] On Behalf Of Erik Iverson > Sent: Monday, August 31, 2009 2:17 PM > To: r-help@r-project.org > Subject: [R] Offtopic, HT vs. HH in coin flips > > Dear R-help, > > Could someone please try to explain this paradox to me? What > is more likely to show up first in a string of coin tosses, > "Heads then Tails", or "Heads then Heads"? > > ##generate 2500 strings of random coin flips > ht <- replicate(2500, > paste(sample(c("H", "T"), 100, replace = TRUE), > collapse = "")) > > ## find first occurrence of HT > mean(regexpr("HT", ht))+1 #mean of HT position, 4 > > ## find first occurrence of HH > mean(regexpr("HH", ht))+1 #mean of HH position, 6 > > FYI, this is not homework, I have not been in school in > years. I saw a similar problem posed in a blog post on the > Revolutions R blog, and although I believe the answer, I'm > having a hard time figuring out why this should be? > > Thanks, > Erik Iverson > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide > http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.