For those interested, the original puzzle was detailed in a TED presentation by Peter Donnelly, which you can see at blog.revolution-computing.com here: http://bit.ly/2l0ZwS The original problem was: which coin-toss sequence do you expect to see first, HTH or HTT? Peter's explanation comes at about 5:00 in the video, and for a lay audience it's great. (The champagne metaphor is wonderful.)
I'm loving the R simulations of this and related problems in this thread! Cheers, # David On Mon, Aug 31, 2009 at 1:55 PM, William Dunlap <wdun...@tibco.com> wrote: > 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. > -- David M Smith <da...@revolution-computing.com> Director of Community, REvolution Computing www.revolution-computing.com Tel: +1 (206) 577-4778 x3203 (San Francisco, USA) Check out our upcoming events schedule at www.revolution-computing.com/events [[alternative HTML version deleted]] ______________________________________________ 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.
